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ELEMENTS 

OF  THE 

THEORY  OF  FUNCTIONS    >V/ 

OF    A  ^       ^    Ak 

COMPLEX    VARIABLE  ^'^i^ 

WITH 
ESPECIAL  EEFERENCE   TO  THE  METHODS  OF  RIEMANN 


DR.   H.  I^UREGE 

liATE   PROFESSOR  IN   THE  UNIVERSITY   OF  FRAGUK 

AUTHORIZED  TRANSLATION  FROM  THE  FOURTH  GERMAN  EDITION 

BY 

GEORGE    EGBERT    FISHER,  M.A.,  Ph.D. 

ASSISTANT  PROFESSOR   OF   MATHEMATICS   IN   TIIF,   UNIVERSITY   OF    PENNSYLVANIA 


ISAAC   J.   SCHWATT,   Ph.D. 

INSTRUCTOR   IN    MATHEMATICS   IN   THE   UNIVERSITY   OF   PENNSYLVANIA 


PHILADELPHIA 

G.   E.   FISHER  AND  I.   J.    SCHWATT 

1896 


Copyright,  1896,  by 
G.  E.  FISHER  AND  I.  J.  SCHWATT. 


NorfaoolJ  33rfg8 

J.  S.  Cuahing  &  Co.  —  Berwick  &  Smith 

Norwood  Mass.  U.S.A. 


£nelnMilng& 
Mathematical 

Sciences 

Liftrarv 

a>3>( 


e 


TRANSLATORS'   NOTE. 

We  desire  to  express  our  indebtedness  to  Professor  James 

McMahon,  Cornell  University,  for  his  kindness  in  reading  the 

translation,  and  for  his  valuable  suggestions.     Also  to  Messrs. 

J.  S.  Gushing  &  Co.,  Norwood,  Mass.,  for  the  typographical 

excellence  of  the  book. 

G.  E.  F. 
I.    J.   S. 
University  of  Pennsylvania, 

Philadelphia,  December,  1895. 


FROM  THE  AUTHOR'S  PREFACE  TO  THE  THIRD 
EDITION. 

Numerous  additions  have  been  made  to  Section  IX.,  which 
treats  of  multiply  connected  surfaces.  If  Riemann's  fundar 
mental  proposition  on  these  surfaces  be  enunciated  in  such 
a  form  that  merely  simply  connected  pieces  are  formed  by 
both  modes  of  resolution,  —  as  is  ordinarily,  and  was  also  in 
§  49,  the  case,  —  then  it  must  be  supplemented  for  further 
applications.  Such  supplementary  matter  was  given  in  §  52 
in  the  classification  of  surfaces,  and  in  §  53,  V.  But  if  we 
express  the  fundamental  proposition  in  the  form  in  which 
Riemann  originally  established  it,  in  which  merely  simply 
connected  pieces  are  formed  by  only  one  mode  of  resolution, 
while  the  pieces  resulting  from  the  other  mode  of  resolution 
may  or  may  not  be  simply  connected,  then  all  difficulties  are 
obviated,  and  the  conclusions  follow  immediately  without 
requiring  further  expedients. 

This  was  shown  in  a  supplementary  note  at  the  end  of 
the  book. 


AUTHOR'S  PREFACE  TO  THE   FOURTH  EDITION. 

In  the  present  new  edition  only  slight  changes  are  made, 
consisting  of  brief  additions,  more  numerous  examples,  differ- 
ent modes  of  expression,  and  the  like. 

In  reference  to  the  above  extract  from  the  preface  of  the 

preceding  edition,  I  have  asked  myself  the  question,  whether 

I  should  not  from  the  beginning  adopt  the  original  Riemann 

enunciation  of  the  fundamental  proposition  instead  of  that 

which  is  given  in  §  49.     Nevertheless,  I  have  finally  adhered 

to  the  previous  arrangement,  because  I  think  that  in  this  way 

the  difference  between  the  two  enunciations  is  made  more 

prominent,  and  the  advantages  of  the  Riemann  enunciation 

are  more  distinctly  emphasized. 

H.  DUREGE. 
Prague,  April,  1893. 


CONTENTS. 


PASS 

Inthodtjction 1 


SECTION  I. 

GEOMETRIC    REPRESENTATION    OF    IMAGINARY    QUANTITIES. 

§  1.  A  complex  quantity  x  +  iy  or  r(cos  0  +  i  sin  4>)  is  represented 
by  a  point  in  a  plane,  of  which  x  and  y  are  the  rectangular, 
r  and  <p  the  polar  co-ordinates.  A  straight  line  is  deter- 
mined in  length  and  direction  by  a  complex  quantity. 
Direction-coefficient 12 

§  2.     Construction  of  the  first  four  algebraic  operations      ....       15 

1.  Addition 15 

2.  Subtraction.     Transference  of  the  origin     ....       16 

3.  Multiplication 18 

4.  Division.    Applications 20 

§  3.     Complex  variables.     They  can  describe  difierent  paths  between 

two  points.    Direction  of  increasing  angles 22 

SECTION  II. 

FUNCTIONS    OF    A   COMPLEX   VARIABLE   IN    GENERAL. 

§  4.     The  correspondence  of  the  values  of  the  variable  and  of  the 

function  is  the  most  essential  characteristic  of  a  function      25 
§  5.     Conditions  under  which  to  =  m  -f-  i»  is  a  function  of  z  =  x  +  iy      27 

§  6.     The  derivative  —  is  independent  oi  dz 29 

dz 

§  7.  If  10  be  a  function  of  z,  the  system  of  to-points  is  similar  in  its 
infinitesimal  elements  to  the  system  of  ^-points.  Con- 
formal  representation.     Conformation 33 


via  CONTENTS. 

SECTION  III. 

MULTIFORM   FUNCTIONS. 

PAG^ 

§  8.     The  value  of  a  multiform  function  depends  upon  the  path 

described  by  the  variable.    Branch-points 37 

§  9.     Two  paths  lead  to  different  values  of  a  function  only  when 

they  enclose  a  branch-point 43 

§  10.     Examples : 

(1)  V^,  (2)  (z-l)V~z,  (3)  ^'^,  (4)  l^+VrTc. 

Cyclical  interchange  of  function-values 47 

§  11.    Introduction  of  Riemann  surfaces  which  cover  the  plane 

ra-fold.    Branch-cuts 55 

§  12.     Proof  that  this  mode  of  representation  of  K-valued  functions 

is  conformal 61 

§  13.     Continuous  and  discontinuous  passage  over  the  branch-cuts. 

Simple  branch-points  and  winding-points  of  higher  orders      68 

§  14.  The  infinite  value  of  z.  Surfaces  closed  at  infinity  (Riemann 
spherical  surfaces).  Proof  that  the  formation  of  a  Rie- 
mann spherical  surface  for  a  given  algebraic  function  is 
always  possible.  Examples  of  different  arrangements  of 
branch-cuts 71 

§  15.    Every  rational  function  of  w  and  z  is  like-branched  with  w   .      80 

SECTION   IV. 

INTEGRALS   WITH   COMPLEX   VARIABLES. 

§  16.     Definition  of  an  integral.    Dependence  of  the  same  upon  the 

path  of  integration 80 

§17.    The   surface   integral    j  f  ( -^  —  "f- )  <^a^y  is  equal  to  the 

linear  integral  f  (Pdx  -f  Qdy)  extended  over  the  boun- 
dary.   Positive  boundary-direction 84 

§  18.  If  Pdx+  Qdy  be  a  complete  differential,  then  \  (Pdx+  Qdy), 
taken  along  the  boundary  of  a  surface  in  which  P  and  Q 
are  finite  and  continuous,  is  zero.  Also,  if{z)dz  =  0,  if 
this  integral  be  taken  along  the  boundary  of  a  surface 
in  which  f(z)  is  finite  and  continuous.  Importance  of 
simply  connected  surfaces 90 

§  19.  The  value  of  a  boundary  integral  does  not  change  when  pieces 
in  which /(«)  is  finite  and  continuous  are  added  to  or  sub- 


CONTENTS.  IX 

PAGE 

tracted  from  the  bounded  surface.  A  boundary  integral 
is  equal  to  the  sum  of  the  integrals  taken  along  small 
closed  lines  which  enclose  singly  all  the  points  of  dis- 

continuitj'  contained  in  the  surface 93 

§  20.  If  f(z)  become  infinite  at  the  point  z  =  a  in  such  a  way  that 
lim[(z—a)f(z)]z~a=P,  then  if{z)dz=2Trip,  integrated 
round  a.      Reduction  of  the  values  of  the  integral  to 

closed  lines  round  the  points  of  discontinuity     ....       9(5 

1 

§21.     Integrals  round  a  branch-point  b.    If  we  let  («  —  fc)"  =  f 

and  /(2)  =  0(f'),  then  ^(f)   does  not  have  a  branch- 

m-l 

point  at  the  place  f  =  0.    If  at  least  lim  [{z  —b)  "*  fz]z=b 

be  finite,  then  (f{z)dz  =  0 101 

SECTION  V. 

THE   LOGARITHMIC   AND    EXPONENTIAL   FUNCTIONS. 

§  22.  Definition  and  properties  of  the  logarithmic  function.  Multi- 
plicity of  values  of  the  same        104 

§  23.     The  exponential  function  z  =  e".     Representation  of  the  z- 

surface  on  the  w-surface 109 

SECTION  VI. 

GENERAL    PROPERTIES    OF    FUNCTIONS. 
§  24.    It  <f){z)  be  uniform  and  continuous  in  a  surface  T,  then  for 

every  point  t  in  that  surface  <f>(t)  = (  ^i£iH£^  the 

2  niJ  z  —  t 
integral  to  be  taken  round  the  boundary  of  T.  In  a 
region  in  which  a  function  (p(z)  is  uniform  and  continu- 
ous, its  derivatives  are  also  uniform  and  continuous,  and 
if  we  put  <p(z)  =  t<  +  iv,  then  neither  mod  u  nor  mod  v 
can  have  a  maximum  or  minimum  value  at  any  place 

in  this  region 113 

§  25.  The  domain  of  a  point  a.  A  function  which  is  finite  and 
continuous  at  a  point  a  can  be  represented  by  a  conver- 
gent series  of  ascending  powers  oi  t  —  a  for  all  points  t 
in  the  domain  of  a.  Taylor's  series.  A  function  of  a 
complex  variable  can  be  continued  in  only  one  way ; 
if  it  be  constant  in  any  arbitrary  small  finite  region,  it 
is  constant  everywliere        117 


X  CONTENTS. 

PAGE 

§  26.  Representation  of  a  function  by  a  convergent  series  in  the 
domain  of  a  point  which  is  not  a  branch-point,  but  at 
which  the  function  is  discontinuous 121 

SECTION  VII. 

INFINITE    AND    INFINITESIMAL    VALUES    OF    FUNCTIONS. 

A.   Ftinctions  without  branch-points.     Uniform  functions. 

§  27.  Polar  and  non-polar  discontinuities.  A  uniform  function  at 
a  point  of  discontinuity  of  the  second  kind  must  become 
infinite  and  be  capable  of  assuming  every  arbitrary  value. 
Example 125 

§  28.  A  uniform  function  which  does  not  become  infinite  for  any 
finite  or  infinite  value  of  the  variable  is  a  constant.  A 
uniform  function  which  is  not  a  mere  constant  must 
become  infinite  and  zero,  and  be  capable  of  acquiring 
any  given  value 131 

§  29.  An  infinite  value  of  a  definite  order.  Further  characteriza- 
tion of  the  two  kinds  of  discontinuity 133 

§  30.     An  infinite  value  for  z  =  co        141 

§  31.  A  uniform  function  which  becomes  infinite  only  for  2  =  oo 
and  that  of  a  finite  order  is  an  integral  function.  If,  for 
2  =00 ,  it  become  infinite  of  an  infinitely  high  order,  it 
can  be  developed  in  a  series  of  powers  of  z  converging 
for  all  finite  values  of  the  variable 143 

§  32.  A  uniform  function  which  becomes  infinite  only  a  finite  num- 
ber of  times  is  a  rational  function 144 

§  33.  0(2)  is  determined,  except  as  to  an  additive  constant,  when 
for  each  of  its  points  of  discontinuity  we  are  given  a 
function  which  becomes  discontinuous  just  as  (p(z)  does,     145 

§  34.    An  infinitesimal  or  zero  value       146 

§  35.  If  in  a  given  region  <p(z)  become  zero  of  multiplicity  n  and 
infinite  of  multiplicity  v,  then  i  d  log  (f>(z)  =  2  7ri(?i  —  y), 
the  integral  to  be  taken  round  the  boundary  of  the 
region 147 

§  36.     A  uniform  function  in  the  whole  infinite  extent  of  the  plane 

is  just  as  often  zero  as  it  is  infinite 150 

§  37.  A  uniform  function  is  determined  except  as  to  a  constant 
factor,  when  once  we  know  all  finite  values  for  which  it 
becomes  infinite  and  infinitesimal,  and  also  the  order  of 
the  infinite  or  infinitesimal  value  for  each 163 


CONTENTS.  xi 


B.   Functions  with  branch-points.     Algebraic  functions. 

PAGE 

§  38.     The  infinite  value  of  an  algebraic  function.      It  becomes 

infinite  of  a  fractional  order  at  a  branch-point   ....     155 

§  39.     Behavior  of  the  derived  function  at  a  branch-point  ....     160 

§  40.     Representation    of   the   surface   in   the   neighborhood   of   a 

branch-point 167 

§  41.  An  n-valued  function,  which  becomes  infinite  of  multiplicity 
m,  is  the  root  of  an  algebraic  equation  between  lo  and  z, 
of  the  7ith  degree  with  regard  to  w,  the  coefiicients  of 
which  are  integral  functions  of  z  at  most  of  the  mth 
degree 169 


SECTION  VIII. 

INTEGRALS. 

A.   Integrals  taken  along  closed  lines. 

§  42.  The  integral  \f(z)dz,  taken  round  a  point  of  discontinuity 
about  which  the  ^-surf  ace  winds  m  times,  and  at  which 
f(z)  becomes  infinite  of  a  finite  order,  has  a  value  differ- 
ent from  zero,  when,  and  only  when,  the  term  which 
becomes  infinite  of  the  first  order  is  present  in  the  ex- 
pression defining  the  nature  of  the  infinite  value  of  f(z) ; 
and  this  value  is  equal  to  2  miri  times  the  coefficient  of 
this  term 172 

§  43.  Closed  lines  round  the  point  at  infinity.  The  integral  taken 
along  such  a  line  depends  upon  the  nature  of  the  function 
22/(2). 175 

B.   Integrals  along  open  lines.     Indefinite  integrals. 

§  44.  Behavior  of  the  integral  of  an  algebraic  function  <l>(_z),  when 
the  upper  limit  acquires  a  value  for  which  <p(z)  becomes 
infinite.     Logarithmic  infinity 180 

§  45.     Behavior  of  the  integral  when  the  upper  limit  tends  towards 

infinity.    Examples 182 


xii  CONTENTS. 

SECTION   IX. 

SIMPLY    AND    MULTIPLY    CONNECTED    SURFACES. 

PAGE 

§  46.  Definition.  Criterion  for  determining  whether  a  closed  line 
forms  by  itself  alone  the  complete  boundary  of  a  region. 

Examples 184 

§  47.     Cross-cuts 189 

§  48.     Preliminary  propositions 192 

§  49.     Riemann's  fundamental  proposition 196 

§  50.    Digression  on  line-systems 206 

§  51.     Lippich's  proposition.    Another  proof  of  the  fundamental 

proposition        209 

§  52.  Classification  of  surfaces  according  to  the  order  of  their  con- 
nection     212 

§  53.     Various  propositions 215 

§  54.  Determination  of  the  order  of  connection  for  a  closed  Rie- 
mann  surface  which  possesses  no  boundary-lines,  but 

only  a  boundary-point 221 

§  55.     The  same  for  a  Riemann  surface  which  is  extended  over  a 

finite  part  of  the  plane 224 

§  56.     The  same  for  an  arbitrary  Riemann  surface  which  possesses 

boundary-lines 232 

§  57.  Extension  of  the  Eulerian  relation  between  the  number  of 
corners,  edges,  and  faces  of  a  body  bounded  by  plane 
surfaces  when  it  has  an  arbitrary  form 242 

SECTION  X. 

MODULI    OF    PERIODICITY. 

§  58.  Examination  of  a  function  defined  by  an  integral  in  a  multi- 
ply connected  surface.  On  crossing  a  cross-cut  the  func- 
tion changes  abruptly  by  a  quantity  which  is  constant 
along  the  cross-cut.  Multiformity  of  the  functions  defined 
by  integrals.    The  inverse  functions  are  periodic    .     .     .     246 

§  59.  Extension  for  the  case  in  which  previous  cross-cuts  are  divided 
into  segments  by  subsequent  cross-cuts.  The  number  of 
independent  moduli  of  periodicity  is  equal  to  the  number 
of  cross-cuts 251 

§  60.  More  rigorous  determination  of  the  points  which  must  be 
excluded  from  the  surface  in  the  examination  of  a  func- 
tion defined  by  an  integral,  and  which  must  not  be  ex- 
cluded       256 


CONTENTS.  xiii  A    ^ 

§  61.     Examples  page   ^  ^      ^ 

1.  The  logarithm 25ay^      ' 

2.  The  inverse  tangent 259     -.^  ,^» 

3.  The  inverse  sine 266  ^   ^ 

4.  The  elliptic  integral 2'Ltf^  J^ 

/* 

Supplementary    note    to    Riemann's   fundamental   proposition    on 

multiply  connected  surfaces 286 


^ 
W 


ELEMENTS 


OF  THE 


THEORY   OF   FUNCTIONS   OF   A 
COMPLEX   VARIABLE. 


it*io 


INTRODUCTION. 

To  follow  the  gradual  development  of  the  theory  of  imagi- 
nary quantities  is  especially  interesting,  for  the  reason  that 
we  can  clearly  perceive  with  what  difficulties  is  attended  the 
introduction  of  ideas,  either  not  at  all  known  before,  or  at 
least  not  sufficiently  current.  The  times  at  which  negative, 
fr3,ctional  and  irrational  quantities  were  introduced  into 
mathematics  are  so  far  removed  from  us,  that  we  can  form 
no  adequate  conception  of  the  difficulties  which  the  intro- 
duction of  those  quantities  may  have  encountered.  Moreover, 
the  knowledge  of  the  nature  of  imaginary  quantities  has 
helped  us  to  a  better  understanding  of  negative,  fractional  and 
irrational  quantities,  a  common  bond  closely  uniting  them  all. 

Among  the  older  mathematicians,  the  view  almost  univer- 
sally prevailed  that  imaginary  quantities  were  impossible. 
In  glancing  over  the  earlier  mathematical  writings,  we  meet 
with  the  statement  again  and  again  that  the  occurrence  of 
imaginary  quantities  has  no  other  significance  than  to  prove 
the  impossibility  or  insolubility  of  a  problem,  that  these 
quantities  have  no  meaning,  but  may  sometimes  be  profitably 
employed,  the  form  of  the  results  being  then  merely  symboli- 

1 


2  THEORY  OF  FUNCTIONS. 

cal.  In  this  connection  it  is  interesting  to  observe  the  de- 
velopment of  Cauchy's  process.  This  great  mathematician, 
together  with  the  "Princeps  mathematicorum,"  Gauss,  who 
had  first,  and  probably  very  early,  recognized  the  great  impor- 
tance of  imaginary  quantities  in  all  parts  of  mathematics,  may 
be  considered  the  joint-creator  of  the  theory  of  functions  of 
imaginary  variables.  Yet,  both  in  his  Algebraical  Analysis 
and  also  in  the  Exercises  of  the  year  1844,  he  still  followed 
entirely  the  views  of  the  older  mathematicians.  In  one  place 
we  read  ^:  "Toute  equation  imaginaire  n'est  autre  chose  que 
la  representation  symbolique  de  deux  equations  entre  quanti- 
tes  reelles.  L'emploi  des  expressions  imaginaires,  en  per- 
mettant  de  remplacer  deux  equations  par  une  seule,  offre 
souvent  le  moyen  de  semplifier  les  calculs  et  d'ecrire  sous  une 
forme  abregee  des  resultats  fort  compliques.  Tel  est  mSme 
le  motif  principal  pour  lequel  on  doit  continuer  a  se  servir  de 
ces  expressions,  qui  prises  a  la  lettre  et  interpretees  d'apres 
les  conventions  generalement  etablies,  ne  signifient  rien  et 
n'ont  pas  de  sens.  Le  signe  V—  1  n'est  en  quelque  sorte  qu'un 
outil,  un  instrument  de  calcul,  qui  peut-etre  employe  avec 
succes  dans  un  grand  nombre  de  cas  pour  rendre  beaucoup 
plus  simples  non-seulement  les  formules  analytiques,  mais 
encore  les  methodes  a  I'aide  desquelles  on  parvient  a  les 
etablir." 

These  words  indicate  very  clearly  the  standpoint  of  the 
older  mathematicians,  which,  as  may  be  seen,  was  still  main- 
tained by  some  at  a  much  later  period.  In  one  only  of  the 
mathematical  branches  have  imaginary  quantities  always  been 
recognized,  namely,  in  the  theory  of  algebraical  equations; 
for  here  it  was  far  too  important  to  consider  all  the  roots 
together,  for  the  imaginary  state  of  any  of  the  latter  to  inter- 
rupt the  investigations.  Nevertheless,  individual  men,  as 
de  Moivre,  Bernoulli,  the  two  Fagnano,  d'Alembert  and  Euler, 
who  seemed  to  turn   to   imaginary  quantities  with  especial 

^Cauchy,  Exercises  cf  analyse  et  de  physique  mathematique,  Tome  III. 
p.  361. 


INTRODUCTION.  3 

predilection,  gradually  discovered  the  distinguishing  proper- 
ties inherent  in  these  quantities,  and  more  and  more  developed 
their  theory.  Still,  as  a  whole,  these  investigations  were 
looked  upon  rather  as  scientific  pastimes,  as  mere  curiosities, 
and  were  held  to  be  of  value  only  in  so  far  as  they  lent  them- 
selves as  aids  to  other  investigations.  And  there  have  not 
been  wanting  those  who  opposed  the  employment  of  imaginary 
quantities  altogether,  on  account  of  their  supposed  impossi- 
bility.i 

The  opinion  that  imaginary  quantities  are  impossible  has 
its  true  origin  in  mistaken  ideas  of  the  nature  of  negative, 
fractional  and  irrational  quantities.  For  the  application  of 
these  mathematical  ideas  to  geometry,  mechanics,  physics, 
and  partially  even  to  civic  life,  presenting  itself  so  readily 
and  so  spontaneously,  and  in  many  cases  no  doubt  even 
giving  rise  to  some  investigation  of  these  quantities,  it  came 
to  be  thought  that  in  some  one  of  these  applications  should 
be  found  the  true  nature  of  such  ideas  and  their  true  posi- 
tion in  the  field  of  mathematics.  Now,  in  the  case  of  imagi- 
nary quantities,  such  an  application  did  not  readily  present 
itself,  and  owing  to  insufficient  knowledge  of  the  same  it  was 
thought  that  they  should  be  relegated  to  the  realm  of  impossi- 
bility and  their  existence  be  doubted. 

But  thereby  it  was  overlooked  that  pure  mathematics,  the 
science  of  addition,  however  important  may  be  its  applications, 
has  in  itself  nothing  to  do  with  the  latter ;  that  its  ideas,  once 
introduced  by  complete  and  consistent  definitions,  have  their 
existence  based  upon  these  definitions,  and  that  its  principles 
are  equally  true,  whether  or  not  they  admit  of  any  applica- 
tions. Whether  and  when  this  or  that  principle  Avill  find  an 
application  cannot  always  be  determined  in  advance,  and  the 

"lAussi  a-t-on  vu  quelques  gSometres  d'un  rang  distingufi  ne  point 
gouter  ce  genre  de  calcul,  non  qu'ils  doutassent  de  la  justesse  de  son 
rfisultat,  mais  parce  qu'il  paraissait  y  avoir  une  sorte  d'inconvenance  h. 
employer  des  expressions  de  ce  genre  qui  n'ont  jamais  servi  qu'a  annon- 
cer  une  absurdity  dans  I'^nonc^  d'un  problgme."  —  Montucla,  Histoire 
des  Mathernatiques,  Tome  III.  p.  283. 


4  THEOBY  OF  FUNCTIONS. 

present  time  especially  is  rich  enough  in  instances  in  which 
the  most  important  applications  —  even  those  of  far-reaching 
influence  on  the  life  of  nations  —  have  sprung  from  principles, 
at  the  discovery  of  which  there  was  certainly  no  suggestion  of 
such  results.  But  so  firm  had  the  belief  in  the  impossibility 
of  imaginary  quantities  gradually  become  that,  when  the  idea 
of  representing  them  geometrically  ^  first  arose  in  the  middle  of 
the  last  century,  from  the  supposed  impossibility  of  the  same, 
was  inferred  conversely  the  impossibility  of  representing  them 
geometrically.^ 

To  understand  the  position  which  imaginary  quantities 
occupy  in  the  field  of  pure  mathematics,  and  to  recognize  that 
they  are  to  be  put  upon  precisely  the  same  footing  as  negative, 
fractional  and  irrational  quantities,  we  must  go  back  some- 
what in  our  considerations. 

The  first  mathematical  ideas  proceeding  immediately  from 
the  fundamental  operation  of  mathematics,  i.e.,  addition,  are 
those  which,  according  to  the  present  way  of  speaking,  are 
called  positive  integers. 

If  from  addition  we  next  pass  to  its  opposite,  subtraction,  it 
soon  becomes  necessary  to  introduce  new  mathematical  con- 
cepts. For,  as  soon  as  the  problem  arises  to  subtract  a  greater 
number  from  a  less,  it  can  no  longer  be  solved  by  means  of 
positive  integers.  From  the  standpoint  in  which  we  deal  with 
only  positive  integers,  we  have  therefore  the  alternative,  either 
to  declare  such  a  problem  impossible,  insoluble,  and  thus  to 

1  On  the  history  concerning  the  geometrical  representation  of  imaginary 
quantities,  compare  Hankel,  Theorie  der  complexen  Zahlensysteme, 
Leipzig,  1867,  S.  81.  It  deserves  to  be  noted  that  Abel  and  Jacobi,  in 
opposition  to  tlie  view  that  only  a  geometrical  representation  could  secure 
for  imaginary  quantities  a  real  existence,  already  made  unlimited  use  of 
imaginary  quantities  in  their  first  investigations  on  elliptic  functions,  and 
this  at  a  time  when  that  representation  was  all  but  unknown.  Fully 
conscious  of  how  essential  the  consideration  of  imaginary  quantities  was, 
and  how  incomplete  their  investigations  would  remain  without 'them, 
they  disregarded  entirely  the  question  of  their  possibility  or  impossibility. 

2Foncenex,  "Reflexions  sur  les  quantit^s  imaginaires,"  Miscellanea 
Taurinensia,  Tome  I.  p.  122. 


INTRODUCTION.  5 

piit  a  stop  to  all  further  progress  of  the  science  in  this  direc- 
tion; or,  on  the  other  hand,  to  render  the  solution  of  the 
problem  possible  by  introducing  as  new  concepts  such  mathe- 
matical ideas  as  enable  us  to  solve  the  problem.  In  this  way 
negative  quantities  at  first  arise  through  subtraction  as  the 
differences  of  positive  integers,  of  which  the  subtrahends  are 
greater  than  the  minuends.  Their  existence  and  meaning 
for  pure  mathematics,  then,  is  not  based  upon  the  opposition 
between  right  and  left,  forward  and  backward,  affirmation  and 
negation,  debit  and  credit,  or  upon  any  other  of  their  various 
applications,  but  solely  upon  the  definitions  by  which  they 
were  introduced. 

ISTow,  although  the  idea  of  impossibility  is  not  at  all  con- 
tained in  our  conceptions  of  negative  quantities,  it  may  happen 
that  the  occurrence  of  negative  quantities  indicates  the  impos- 
sibility or  insolubility  of  a  problem,  namely,  when  the  nature 
of  the  problem  necessarily  requires  positive  quantities  for  its 
solution.  If,  for  instance,  the  following  problem  be  given: 
Six  balls  are  to  be  distributed  in  two  urns,  so  that  one  shall 
contain  eight  more  than  the  other;  then  the  following  purely 
mathematical  problem  is  contained  in  it :  to  find  two  numbers 
of  which  the  siim  is  equal  to  six  and  the  difference  to  eight. 
Now,  if  it  merely  be  desired  that  the  numbers  shall  be  mathe- 
matical concepts  without  limiting  them  to  a  special  kind,  and 
if,  moreover,  the  conception  of  negative  quantities  has  been 
fixed  beforehand  by  defining  them,  the  solubility  of  the  purely 
mathematical  problem  is  quite  obvious  —  the  positive  number  7 
and  the  negative  number  —  1  are  the  quantities  which  satisfy 
the  problem.  Nevertheless,  it  is  impossible  to  solve  the 
problem  originally  set,  for  it  requires  that  each  of  the  num- 
bers sought  shall  stand  for  a  quantity,  and  therefore  neces- 
sarily be  positive.  If  the  impossibility  were  not  so  obvious  as 
it  is  in  this  simple  example,  the  occurrence  of  the  negative 
num'ber  —  1  would  show  conclusively  the  insolubility  of  the 
problem. 

Exactly  the  same  conditions  arise  in  every  other  inverse 
operation.     The  next  inverse  operation  is  division.     If  we  set 


6  THEORY  OF  FUNCTIONS. 

the  problem  to  divide  a  whole  number  by  another  which  is 
not  a  factor  of  the  first,  there  arises  the  impossibility  of 
solving  this  problem  by  positive  or  negative  integers.  The 
progress  of  the  science  therefore  again  requires  the  possibility 
of  the  solution  to  be  brought  about  by  introducing  and  defining 
the  quantities  necessary  to  that  end.  Here  these  new  con- 
cepts are  rational  fractions.  But  here,  too,  the  case  may  occur 
that  the  appearance  of  such  quantities  proves  the  impossibility 
of  solving  a  particular  problem ;  and  again,  as  before,  when  by 
the  nature  of  the  problem  it  does  not  admit  of  a  solution  in 
terms  of  the  new  concepts.  Take  as  an  example  the  following 
problem :  A  wheel  in  a  machine  or  clock  work,  which  has  100 
cogs  and  revolves  once  a  minute,  is  to  set  directly  in  motion 
another  wheel,  so  that  the  latter  shall  make  12  revolutions  in  a 
minute ;  how  many  cogs  must  we  give  to  the  second  wheel  ? 
In  this  case  the  underlying  purely  mathematical  problem  con- 
sists merely  in  dividing  100  by  12;  and  if  the  definition  of 
fractions  has  once  been  given,  the  solution  presents  no  diffi- 
culty, the  result  being  8^.  But  the  occurrence  of  this  fraction 
proves  at  once  the  impossibility  of  solving  the  problem  origi- 
nally proposed,  as  the  number  of  cogs  on  the  second  wheel  to 
be  determined  must  be  an  integer. 

The  third  inverse  operation  is  the  extraction  of  roots. 

Given  Va  =  x, 

in  which  n  denotes  a  positive  integer ;  the  problem  to  find  a 
quantity  x  satisfying  this  equation  can  no  longer  be  solved 
in  terms  of  whole  numbers  or  rational  fractions,  as  soon  as 
a  is  not  the  ?ith  power  of  such  a  quantity.  In  this  case 
therefore  the  necessity  again  arises  of  rendering  the  problem 
soluble  by  the  introduction  of  new  concepts.  Now,  if  either 
a  be  positive,  or  in  case  a  is  negative,  if  n  be  an  odd  number, 
the  new  concepts  to  be  introduced  are  irrational  quantities ;  but 
if  a  be  negative,  and  n  at  the  same  time  an  even  number,  the 
new  concepts  to  be  introduced  are  imaginary  quantities.  Now 
it  is  no  more  an  impossibility  to  define  these  latter  than  to 


INTRODUCTION.  7 

define  irrational  quantities,  or,  to  go  back  still  farther,  than  to 
define  rational  fractions  and  negative  quantities,  for  in  none 
of  these  definitions  do  we  meet  with  any  inherent  incon- 
sistencies. Should  such  occur,  should  properties  be  put  in 
combination  with  one  another  which  we  can  prove  to  be  incon- 
sistent, then,  it  must  be  admitted,  we  should  have  actually 
to  deal  with  an  impossibility.  Gauss  ^  adduces  as  an  example 
of  such  an  impossibility  a  plane  rectangular  equilateral  tri- 
angle. And  indeed  it  can  be  proved  that  a  plane  equilateral 
triangle  cannot  at  the  same  time  be  rectangular.  Something 
impossible  would  therefore  actually  be  proposed.  If  now,  in 
fact,  the  occurrence  of  negative  quantities,  or  of  fractions, 
indicate  sometimes  the  impossibility  of  particular  problems, 
it  is  easily  conceivable  that  such  an  impossibility  can  also 
be  proved  by  means  of  imaginary  quantities,  as  in  the  follow- 
ing example:  A  given  straight  line  two  units  long  is  to  be 
divided  into  two  such  parts,  that  the  rectangle  formed  by 
them  shall  have  the  area  4.  The  purely  mathematical  con- 
tent of  this  problem  is  to  find  two  numbers  of  which  the  sum 
equals  2  and  the  product  4.  If  now  it  be  required  merely 
that  these  numbers  shall  be  mathematical  quantities,  without 
specifying  the  particular  kind,  then,  the  definition  of  imagi- 
nary quantities  having  once  been  given,  the  solution  presents 
no  difiiculty.  It  leads  to  the  solution  of  the  quadratic  equa- 
tion, 

a^_2a;  +  4  =  0, 

of  which  the  roots  are  the  imaginary  quantities 


But  if  we  attempt  to  satisfy  the  conditions  of  the  original 
problem,  that  the  quantities  sought  shall  represent  parts  of  a 
straight  line  and  hence  be  real  quantities,  it  is  impossible 
to  solve  the  problem,  because  the  greatest  rectangle  formed 

1 "  Demonstratio  nova  theorematis  omnem  functionem  algebraicam 
rationalem  integram  unius  variabilis  in  factores  reales  primi  vel  secundi 
gradus  resolvi  posse." — Inaug.  Diss.  p.  4,  Note. 


8  THEORY  OF  FUNCTIONS. 

by  two  parts  of  the  line  2  has  the  area  1,  and  therefore  none 
can  have  the  area  4;  and  this  impossibility  is  indicated  in 
this  case  by  the  occurrence  of  imaginary  quantities.  Mon- 
tucla^  has  chosen  this  very  example  in  support  of  his  view 
that  the  meaning  and  origin  of  imaginary  quantities  are  to 
be  looked  for  altogether  in  the  impossibility  of  a  problem, 
because  these  quantities  occur  when  a  problem  is  given  which 
contains  an  impossible  or  absurd  condition.  We  have  already 
seen  that  exactly  the  same  can  be  affirmed  of  negative  quanti- 
ties and  fractions,  and  the  words :  "  Ainsi  toutes  les  fois  que 
la  resolution  d'un  probleme  conduit  a  de  semblables  expres- 
sions et  que  parmi  les  differentes  valeurs  de  I'inconnue  il  n'y 
en  a  que  de  telles,  le  probleme,  ou  pour  mieux  dire,  ce  qu'on 
demande  est  impossible,"  and  further  on,  "  Le  probleme,  qui 
conduirait  a  une  pareille  equation,  serait  impossible  ou  ne 
presenterait  qu'une  demande  absurde,"  can  be  applied  almost 
literally  to  the  two  examples  adduced  above,  in  which  the 
impossibility  of  the  problem  was  indicated  by  a  negative 
number  and  by  a  fraction  respectively. 

It  is  evident  from  the  foregoing  considerations  that  imagi- 
nary, irrational,  rational-fractional  and  negative  quantities, 
have  all  a  common  mode  of  origin,  namely,  by  means  of 
inverse  operations,  in  which  their  introduction  is  rendered 
necessary  by  the  further  progress  of  the  science.  They  all 
have  their  existence  based  upon  their  definitions,  no  one  of 
which  includes  anything  impossible ;  but  it  may  happen  that 
the  occurrence  of  each  of  them  proves  the  impossibility  of 
solving  a  given  problem,  on  account  of  the  peculiar  character 
of  the  same. 

Before  we  take  up  the  subject  proper,  some  remarks  on  the 
calculations  by  means  of  imaginary  quantities  may  be  per- 
mitted. Here,  too,  we  can  start  from  quantities  related  to 
them.  Every  time  a  new  concept  is  introduced  into  mathe- 
matics, it  is  in  many  respects  absolutely  a  matter  of  choice  in 
what  way  the  operations  upon  which  the   former   concepts 

1  Histoire  des  Mathematiques,  Tome  III.  p.  27. 


INTRODUCTION.  9 

depend  shall  be  transferred  to  the  new.  For  instance,  after 
the  definition  of  powers  with  positive  integral  exponents  has 
been  derived  from  the  repeated  multiplication  of  a  quantity  by 
itself,  the  question  arises  as  to  what  is  to  be  understood  by  a 
power  with  a  negative  exponent.  In  itself  the  answer  is  abso- 
lutely a  matter  of  choice,  for  there  is  nothing  which  compels 
us  to  understand  by  it  one  thing  and  no  other.  But  if  in  this 
and  all  similar  cases  we  had  proceeded  quite  arbitrarily,  and 
had  not  been  guided  by  any  definite  principle,  the  structure  of 
mathematics  would  surely  have  assumed  a  strange  form,  and 
the  survey  of  it  enormous  difficulty.  Mathematics  owes  its 
external  consistency  and  the  harmonious  agreement  of  all  its 
parts  to  the  adherence  to  the  principle  that  every  time  a 
newly  introduced  concept  depends  upon  operations  previously 
employed,  the  propositions  holding  for  these  operations  are 
assumed  to  be  valid  still  when  they  are  applied  to  the  new 
concepts.  This  assumption,  arbitrary  in  itself,  it  is  permissible 
to  make,  as  long  as  no  inconsistencies  result  from  it.^  Now 
when  this  principle  is  adhered  to,  the  definitions  which  have 
been  discussed  above  are  no  longer  arbitrary,  but  follow  as 
necessary  results  of  that  principle.  In  the  case  of  powers,  for 
instance,  it  is  proved  that  when  m  and  n  are  two  positive 
integers,  and  we  assume  that  m  >  n,  then 

—  =  a'»-» 
a" 

Now  we  arbitrarily  assume  that  this  theorem  remains  true 
also  when  m  < « ;  that  is,  when  m  —  n=p  is  a  negative  num- 
ber ;  and  it  follows  that  we  have  to  put 

a" 

by  which  the  meaning  of  a  power  ^vith  a  negative  exponent  is 
now  definitely  determined. 

1  This  is  the  same  assumption  that  was  called  by  Hankel  the  principle 
of  the  permanence  of  the  formal  laws.  Theorie  der  complexen  Zahlen- 
systeme,  Leipzig,  1867,  S.  11. 


10  THEORY  OF  FUNCTIONS. 

No  further  argument  is  needed  to  prove  that  the  above 
principle  is  of  the  greatest  importance  for  mathematics,  not- 
Avithstanding  the  fact  that  its  assumption  is  by  no  means 
necessary  but  arbitrary. 

We  need  only  realize  how  the  system  of  mathematics  would 
be  constituted,  were  that  principle  not  adhered  to,  in  order  to 
see  at  once  what  distinctions  we  should  be  forced  to  make  at 
each  step,  and  how  cumbersome  would  become  the  methods 
of  proof.  The  generalizations  of  mathematical  principles 
brought  about  by  the  prevalence  of  this  principle  to  the  widest 
extent  explain  also  another  phenomenon  in  the  history  of 
mathematics,  namely,  that  for  a  long  time  the  views  in  regard 
to  the  meaning  of  divergent  series  differed  so  radically.  As  it 
had  been  the  habit  to  accept  all  mathematical  propositions  as 
holding  generally,  it  required  some  time  for  the  conviction 
to  prevail  that  in  the  development  of  series  the  results  hold 
only  under  certain  limiting  conditions,  and  that  in  general  on 
the  introduction  of  infinity  into  mathematics,  the  principle 
stated  above  does  not  admit  of  as  unconditional  applications  as 
before. 

But  in  transferring  mathematical  processes  to  imaginary 
quantities,  the  above  principle  admits  of  the  fullest  applica- 
tion, and  it  has  been  conclusively  proved  that  thereby  no 
inconsistencies  arise.  It  is  not  our  purpose  here  to  repeat  the 
proof;  it  may,  however,  be  mentioned  that  that  principle, 
although  in  other  respects  always  followed,  yet  in  the  case 
of  imaginary  quantities  has  not  always  and  generally  been 
accepted.  As  late  as  Euler's  time  mathematicians  were  not 
yet  unanimous  in  regard  to  the  meaning  of  the  product  of  two 
square  roots  of  negative  quantities.  Euler  himself  taught, 
conformably  with  the  above  principle  and  as  now  generally 
accepted,  that,  if  a  and  b  denote  two  positive  quantities, 

V— a  •  V—  6  =Va6; 

i.e.,  that  the  product   of   these  two   imaginary  quantities   is 
equal  to  a  real  quantity.     But  this  view  was  not  generally 


INTRODUCTION.  11 

accepted,  and  Emerson,  an  English  mathematician,  taught  on 
the  contrary  that  we  are  forced  to  assume  that 

V—  a  ■  V—  b  =  ^—ab, 

because  it  would  be  absurd  to  assume  that  the  product  of  two 
impossible  quantities  should  not  also  be  impossible ;  and 
Hutton  says  in  his  Mathematical  Dictionary  ^  that  in  his  time 
the  views  of  mathematicians  were  about  equally  divided  on 
this  point. 

One  of  the  remarkable  properties  possessed  by  imaginary 
quantities,  is  that  all  can  be  reduced  to  a  single  one,  namely, 
the  V— 1,  for  which  Gauss  has  introduced  the  now  generally 
accepted  letter  i.^  By  means  of  it,  moreover,  we  can  also 
reduce  every  imaginary  quantity  to  the  form 

z  =  x-\-iy, 

in  which  x  and  y  denote  real  quantities.  A  quantity  of  this 
form  Gauss  has  called  a  complex  quantity,^  divesting  this 
term  of  the  general  meaning  in  which  it  had  sometimes  been 
used  before,  and  according  to  which  it  denoted  any  quantity 
composed  of  heterogeneous  parts,  and  employing  the  term  to 
designate  a  special  heterogeneous  compound,  in  which  a  quan- 
tity consists  of  a  real  and  an  imaginary  part  connected  by 
addition. 

The  complex  quantities  comprise  also  the  real  ones,  namely, 
in  the  case  when  the  real  quantity  y  has  the  value  zero.  If, 
on  the  other  hand,  the  other  real  quantity  be  equal  to  zero, 
and  z  therefore  be  of  the  form 

z  =  iy, 

the  complex  quantity  is  called  a  pure  imaginary.  If,  in  the 
quantity  z  =  x-\-  iy,  either  one  or  both  of  the  real  quantities 

1  Hutton,  Mathematical  Dictionai'y,  1796. 

2  The  first  place  in  which  this  notation  is  employed  is  found,  Dis- 
quisitiones  arithmeticae,  Sect.  VII.  Art.  337. 

3"Theoria  residuorum  biquadraticorum,"  Comment,  societatis  Got- 
tingensis,  Vol.  VII.  (ad.  1828-32),  p.  96. 


12  THEORY  OF  FUNCTIONS. 

X  and  y  be  variable^  z  is  called  a  complex  variable.  In  order 
that  this  sha}l  assume  the  value  zero,  it  is  necessary  for  both 
the  real  quantities  x  and  y  to  vanish  simultaneously,  because 
it  is  not  possible  for  the  two  heterogeneous  quantities,  the 
real  x  and  the  imaginary  iy,  mutually  to  cancel  each  other. 
On  the  other  hand,  in  order  that  the  complex  quantity  z  shall 
become  infinitely  large,  it  suffices  if  only  one  of  its  two  real 
components  x  and  y  become  infinitely  large.  Likewise,  another 
interruption  of  continuity  occurs  in  z  as  soon  as  either  one  of 
the  real  quantities  x  and  y  suffers  such  an  interruption.  But 
as  long  as  both  x  and  y  vary  continuously,  z  is  also  called  a 
continuous  variable  complex  quantity. 

Even  the  consideration  of  real  variables  and  their  functions 
is  materially  facilitated  and  rendered  most  intelligible  by  the 
geometrical  representation  of  the  same.  In  a  much  higher 
degree  is  this  the  case  with  complex  variables ;  we  will  there- 
fore first  examine  the  methods  of  graphically  representing 
imaginary  quantities. 


SECTION   I. 


THE   GEOMETRICAL   REPRESENTATIOlSr   OF   IMAGINARY 

QUANTITIES. 

1.  In  order  to  form  a  geometrical  picture  of  a  real  variable, 
we  conceive,  as  is  well  known,  a  point  moving  on  a  straight 
line.  On  this,  which  we  may  call  the  avaxis,  or  also  the 
principal  axis,  we  assume  a  fixed  point  o  (the  origin),  and 
represent  the  value  of  a  variable  quantity  x  by  the  distance 
op  of  a  point  p  on  the  avaxis  from  the  origin  o.  At  the  same 
time  attention  is  paid  to  the  direction  of  the  distance  op  start- 
ing from  0,  a  positive  value  of  x  being  represented  by  a  distance 
op  toward  one  side  (say,  toward  the  right,  if  the  rc-axis  be 
supposed  to  be  horizontal),  a  negative  value  of  a;  by  a  distance 
op  toward  the  opposite  side  (toward  the  left).     When  now  x 


IMAGINABY  QUANTITIES.  13 

changes  its  value,  the  distance  op  also  changes,  the  point 
23  changing  its  position  on  the  avaxis.  We  can  therefore  say, 
either  that  every  value  of  x  determines  the  position  of  a  point 
p  on  the  a>axis,  or  that  it  determines  the  length  of  a  definite 
straight  line  in  either  of  tAvo  directions  exactly  opposite  to 
each  other. 

A  complex  variable  quantity  z  =  x  +  iy  depends  upon  two 
real  variables  x  and  y,  which  are  entirely  independent  of  each 
other.  Hence  for  the  geometrical  representation  of  a  complex 
quantity  a  range  of  one  dimension,  a  straight  line,  will  no 
longer  suffice,  but  a  region  of  two  dimensions,  a  plane,  will  be 
required  for  that  purpose.  The  manner  of  variation  of  a  com- 
plex quantity  can  then  be  represented  by  assuming  that  a  point 
p  of  the  plane  is  determined  by  a  complex  value  z  —  x-\-iy  in 
such  a  way  that  its  rectangular  co-ordinates,  in  reference  to 
two  co-ordinate  axes,  assumed  to  be  fixed  in  the  plane,  have 
the  values  of  the  real  quantities  x  and  y.  In  the  first  place, 
this  method  of  representation  includes  that  of  real  variables, 
for  when  once  z  becomes  real,  and  therefore  y  —  0,  the  repre- 
senting point  p  lies  on  the  avaxis.  Next,  the  co-ordinates  of 
the  point  p  can  vary  independently  of  each  other,  just  as  the 
variables  x  and  y  do,  so  that  the  point  p  can  change  its  posi- 
tion in  the  plane  in  all  directions.  Further,  one  of  the  two 
quantities,  x  and  y,  can  remain  constant,  while  only  the  other 
changes  its  value,  in  which  case  the  point  p  will  describe 
a  line  parallel  to  the  x-  or  y-axis.  Finally  and  conversely,  for 
every  point  in  the  plane  the  corresponding  value  of  z  is  fully 
determined,  since  by  the  position  of  the  point  p  its  two  rec- 
tangular co-ordinates  are  given,  and  therefore  also  the  values 
of  X  and  y. 

Instead  of  determining  the  position  of  the  point  p  repre- 
senting the  quantity  z  by  rectangular  co-ordinates  x  and  y,  we 
can  accomplish  the  same  by  means  of  polar  co-ordinates.  For, 
by  putting 

x  =  r  cos  <^  and  y  =  r  sin  <^, 
we  obtain  z  =  r  (cos  <^  -f-  ?  sin  <f). 


14  THEORY  OF  FUNCTIONS. 

The  real  quantity  r,  which  is  always  to  be  taken  positively, 

and  which  is  called  the  modulus  of  the  complex  variable  z, 
represents  then  the  absolute  length  of 
the  distance  ^  (Fig.  1),  and  <fi,  called 
the  amplitude  or  argument  of  z,  the  in- 
clination of  that  stroke  to  the  principal 
axis.  Hence  we  can  also  say  that  a 
complex  quantity  r(cos  <f>-\-i  sin  <^)  rep- 
resents a  straight  line  in  length  and 
direction,   namely,   a    straight   line    of 

which  the  length  is  equal  to  r,  and  which  forms  an  angle  <^ 

with  the  principal  axis.     The  quantity 

cos  <fi  +  i  sin  <f>, 

which  depends  upon  this  angle  and  therefore  only  upon  the 
direction  of  the  stroke,  is  usually  called  the  direction-coefficient 
of  the  complex  quantity  z. 

Just  as  we  can  express  by  a  real  number  any  limited  straight 
line,  without  regarding  its  direction  and  position  in  the  plane, 
or,  at  most,  taking  into  account  only  directions  exactly  oppo- 
site to  each  other ;  so  we  can  express  by  a  complex  quantity 
a  straight  line  which  is  determined  both  in  length  and  direc- 
tion, but  of  which  the  position  in  the  plane  is  not  important. 
Two  given  limited  straight  lines  in  a  plane  can  actually  differ 
completely  in  three  particulars :  in  length,  direction  and  posi- 
tion, i.e.,  the  position  of  that  point  at  which  the  line  is  assimied 
to  begin.  We  can,  however,  leave  out  of  consideration  two  of 
these  distinguishing  marks,  and  consider  two  distances  as 
equal,  if  they  have  only  equal  lengths ;  this  is  the  case  in  the 
representation  of  distances  by  real  quantities.  But  in  the 
representation  by  complex  quantities,  we  dispense  with  only 
the  third  distinguishing  mark,  namely,  the  position,  and  call 
two  distances  equal  when,  and  only  when,  they  have  equal 
lengths  and  directions. 

Since  the  modulus  of  a  complex  quantity  determines  the 
absolute  length  of  the  straight  line  representing  that  quantity, 
it  is  analogous  to  the  absolute  value  of  a  negative  quantity 


IMAGINABT  QUANTITIES.  15 

and  serves  as  a  measure  in  comparing  complex  quantities  with 
one  another. 

2.  From  the  property  of  complex  quantities  that  a  combina- 
tion of  two  or  more  of  them  by  means  of  mathematical  operations 
always  leads  again  to  a  complex  quantity,  it  follows  that,  if 
given  complex  quantities  be  represented  by  points,  the  result 
of  their  combination  is  capable  of  being  again  represented  by 
a  point.  We  will  now  in  the  following  examine  the  first  four 
algebraical  operations, — addition,  subtraction,  multiplication 
and  division,  —  and  inquire  how  the  points  resulting  from 
these  operations  can  be  found  geometrically.  In  this  the 
complex  quantities,  and  the  points  representing  them,  will 
always  be  designated  by  the  same  letters;  the  origin,  which 
represents  the  value  zero,  will  be  designated  by  o. 


1.   Addition. 

Let  u  =  x-\-iy  and  v  =  x'  +  iy' 

be  two  complex  quantities,  and  let  w  denote  their  sum ;  then 

w  =  u-{-v=(x-\-  x')-\-i(y  -f  y'). 

The  point  w  therefore  has  the  co-ordinates  a; +  a;'  and  y  +  y'. 
It  follows  that  it  is  the  fourth  vertex  of  the  parallelogram 

formed  on  the  sides  ou  and  ov,  or  v w=u+v 

that  by  the  quantity  u  +  v  is  rep- 
resented the  diagonal  ow  of  this  par- 
allelogram in  magnitude  and 
direction  (Fig.  2).    Since  the 
straight  lines  uw  and  ov  are 
equal   and  directly  parallel,  -v 
and  since  therefore  uw  is  like- 
wise represented  by  the  complex  quantity  v,  we  arrive  at  the 
identical  point  w,  if  we  draw  from  the  end-point  u  of  the  first 
line  ou  the  second  line  ov  in  its  given  length  and  direction. 
This  method  of  combination,  or  geometrical  addition  of  straight 


w=u-v 


16  THEORY  OF  FUNCTIONS. 

lines,  has  been  applied  by  Mobius  ^  independently  of  the  con- 
sideration of  imaginary  quantities.  Accordingly,  the  sum  u+v 
is  the  third  side  of  a  triangle,  of  which  the  two  other  sides  are 
represented  by  u  and  v.  Since,  however,  in  every  triangle 
one  side  is  less  than  the  sum  of  the  two  other  sides,  and 
the  lengths  of  the  sides  are  given  by  the  moduli  of  the 
complex  quantities,  the  proposition  follows:  the  modulus 
of  the  sum  of  two  complex  quantities  is  less  than  (or  equal 
to  ^)  the  sum  of  their  moduli : 

mod  (u-\-v)  -^  mod  u  +  mod  v.^ 
The  complex  quantity  z  =  x  -\-  iy  itself  appears  under  the 
form  of  a  sum  of  the  real  quantity  x  and  the  pure  imaginary 
iy,  since  the  former  is  represented  by  a  point  on  the  avaxis, 
the  latter  by  a  point  on  the  2/-axis,  z  is  in  fact  the  fourth 
vertex  of  the  rectangle,  the  sides  of  which  are  formed  by  the 
abscissa  x  and  the  ordinate  y  of  the  point  z.  a 

2.   Subtraction. 

The  subtraction  of  the  numbers  represented  by  two  points 
can  easily  be  deduced  from  the  addition  of  the  same;    for, 

w=ujrv    given      to'  =  w  —  -y, 
it  follows  that 

u  —  v-\-  w'; 


^u 

therefore  the  point  w'  must  be  so 
situated  that  ou  forms  the  diagonal 
'"w=u-v.       Qf  \\^Q  parallelogram   constructed 
on  ov  and   ovr  (Fig.  2).     Conse- 
quently, we   obtain  w^  by   drawing   ow'  equal    and   directly 
parallel  to  the  straight  line  vxi.     Since,  however,  we  pay  no 
attention  to  the  position  of  a  straight  line,  but  only  to  its 

1  Mobius,  "Uber  die  Zusammensetzung  gerader  Linien,"  etc.     CrelWs 
Journ.,  Bd.  28,  S.  1. 

■■2  When  the  moduli  of  u  and  v  are  drawn  in  the  same  direction, 

mod  (u  +  v)  =  mod  u  +  mod  v.     (Translators.) 
8  Mod  z  (modulus  of  z)  is  sometimes  denoted  by  \z\.     (Tr.) 


IMAGINARY  QUANTITIES.  17 

length  and  direction,  the  difference  u  —  v  is  represented  by 
the  straight  line  vu  in  length  and  direction  (namely,  from 
V  to  u).  The  construction  shows  that  u  falls  in  the  middle 
of  the  straight  line  ww'.     But  from 

w  =  u  +  V,    iv'  =u  —  v, 
it  follows  that 

w  -\-w' 
u  =  ^-^ — : 


9/7     I     7/7 

therefore  the   point  — — —  forms  the  mid-point  of  the  line 

joining  the  points  w  and  w'. 

If  the  point  u  coincide  with  the  origin,  i.e.,  if  m  =  0,  then 
w'  =  —V.  In  this  case  a  line  is  to  be  drawn  from  o  equal  in 
length  and  direction  to  the  line  vo;  hence  the  point  —  v  lies 
diametrically  opposite  to  the  point  v,  and  equally  distant  from 
the  origin. 

Subtraction  furnishes  a  means  of  referring  points  to  another 
origin.     For  it  is  evident  that  a  point  z  is  situated  with  refer- 


ence to  a  point  a  exactly  as  2  —  a  is  situated  with  reference  to 
the  origin  (Fig.  3). 
If  we  put  then 

z  —  a  =  r  (cos  <t>  +i  sin  <f>), 

r  denotes  the  distance  az,  and  <{>  the  inclination  of  the  line  az 
to  the  principal  axis.  The  introduction  of  z'  =  z  —  a,  or  the 
substitution  of  z  +  a  for  z,  transfers  therefore  the  origin  to  a 
without,  however,  changing  the  direction  of  the  principal 
axis. 


18 


THEORY  OF  FUNCTIONS. 


3.  MuUiplicaiion. 

We  employ  here  the  expression  of  complex  quantities  in 
terms  of  polar  co-ordinates.     Let 

u  =  r  (cos  4>  +  i  sin  <^)  and  v  =  r'  (cos  <^'  +  i  sin  <^') 

be  represented  by  two  points  by  means  of  polar  co-ordinates, 
and  let  w  be  their  product ;  then 

w  =  u-v  =  rr'  [cos  (^  +  <^')  +i  sin  (</>  +  ^')]. 

Consequently,  the  radius  vector  of  w  forms  with  the  principal 
axis  the  angle  <f>  -\-  cji',  and  its  length  is  equal  to  the  product  of 

the  numbers  r  and  r',  which 
denote  the  lengths  of  the  radii 
vectors  of  u  and  v.  From  this 
it  follows  that  the  position  of 
the  point  w,  or  w  •  v,  depends 
essentially  upon  the  straight 
line  chosen  as  the  unit  of 
length,  while  the  positions  of 
u  -^  V  and  u  —  v  are  indepen- 
dent of  this  unit.  This  is 
quite  in  accordance  with  the 
nature  of  things,  for  if  in  u 
and  V  the  unit  of  length  be 
increased  in  the  ratio  of  1  to  p, 
p  denoting  a  real  number,  the 
radii  vectors  of  w  +  -u  and  u  —  v  are  increased  in  the  same  ratio ; 
the  radius  vector  of  u-v,  however,  is  increased  in  the  ratio 
of  1  to  p^.  Let  us  assume  then  on  the  positive  side  of  the 
principal  axis  a  point  1,  so  situated  that  o  1  is  equal  to  the 
assumed  unit  of  length  (Fig.  4),    Since  then  from  the  equation 

ow  =  r-r' 

we  obtain  the  proportion      1 :  r  =  r' :  oio, 


or 

and  in  addition 


ol:  ou  —  ov:  oio, 
Z  voio  =  Z  1  ou, 


IMAGINARY  QUANTITIES.  19 

the  position  of  the  point  w  is  to  be  constructed  by  making  the 
triangles  vow  and  1  ou  directly  similar.  Instead  of  this,  we 
could,  of  course,  also  make  the  triangles  now  and  1  ov  directly 
similar,  which  analytically  is  manifested  by  the  fact  that  in 
the  product  u-v  the  factors  are  commutative.  From  the 
equation 

w  =  u-v 

we  can  deduce  another  proportion,  namely, 

1  :  u  =  v:  w; 

hence  the  straight  lines  o  1,  ou,  ov,  oio  are  proportional  to  one 
another,  even  if  their  directions  be  considered.  In  connection 
with  the  preceding,  however,  it  follows  that,  when  straight 
lines  are  compared  with  one  another,  not  only  with  regard  to 
length  but  also  with  regard  to  direction,  two  pairs  of  such 
lines  are  proportional,  when,  and  only  when,  they  not  only  are 
proportional  in  length  but  also  in  pairs  include  equal  angles ; 
or,  in  other  words,  when  they  are  the  corresponding  sides  of 
directly  similar  triangles.  Now,  if  we  take  this  requirement 
into  consideration,  the  last  of  the  above  stated  propositions 
serves  to  find  in  the  simplest  manner  which  triangles  have  to 
be  made  similar  to  each  other ;  for,  from  the  proportion  1 :  u 
=  v:w,  it  follows  by  the  insertion  of  the  point  o  that  the 
triangles  1  ou  and  vow  must  be  similar. 

If  in  the  product  U'V  one  of  the  two  factors,  say  v,  be  real, 
and  if  in  this  case  we  denote  it  by  a,  then  the  point  representing 
a  lies  on  the  principal  axis ;  hence  it  follows  from  the  above 
stated  construction  that  the  point  representing  a-u  lies  on  the 
line  ou  and  at  such  a  distance  from  o  that  its  radius  vector  is 
a  times  the  radius  vector  of  u. 

Consequently,  the  geometrical  meaning  of  multiplication  is 
the  following :  if  a  quantity  u  be  multiplied  by  a  real  quantity 
a,  the  radius  vector  of  u  is  merely  increased  in  the  ratio  of 
1  to  a ;  but  if  ic  be  multiplied  by  a  complex  quantity  v,  the 
radius  vector  of  u  is  not  only  increased  in  the  ratio  of  1  to 
mod  V,  but  it  is  also  turned  through  the  angle  of  inclination  of 
V  in  the  direction  in  which  the  arguments  increase. 


20 


THEORY  OF  FUNCTIONS. 


4.  Division. 

This   operation   follows    immediately   from   the    preceding 
results.     For  if 

w'  =  -, 

V 

we  obtain  therefrom  the  proportion 

w'  :1  =  u:v; 

and  hence  we  have  to  make  the  triangles  iv'ol  and  uov  directly 
similar  (Fig.  4).  The  geometrical  performance  of  the  division 
of  w  by  V  therefore  consists  in  changing  the  radius  vector  of  u 
in  the  ratio  of  mod  v  to  1  and,  at  the  same  time,  in  turning  it 
through  the  angle  of  inclination  of  v  in  the  direction  in  which 
the  arguments  decrease. 

We   will   now  apply  the   foregoing  considerations   to   two 
problems  which  will  be  of  use  to  us  later. 

First:  Let  z,  z'  and  a  be  three  given  quantities,  therefore  also 
three  given  points  ;  we  are  so  to  determine  a  fourth  point  w  that 

w=^-^^^     (Fig.  5). 


If  we  put  z'—z=u  and  z—a=v, 
we  first  find  the  points  repre- 
senting u  and  V  by  drawing  ou 
equal  and  parallel  to  zz',  and  ov 
equal   and   parallel  to  az.     We 

then  have  w=  -,  ot  w.l  =u:v: 

V 

hence  we  obtain  w  by  making  the  A  wo  1  similar  to  A  uov. 
From  this  we  can  also  now  deduce  for  the  quantity  w  an 
expression  which  is  derived  from  the  sides  zz'  and  az  and 
the  angle  azz'  of  the  triangle  azz'.  For,  if  this  angle  be  denoted 
by  a,  then 

Zloio  =  Z  vou  =  180°  —  a. 


Fig.  5. 


moreover, 


—     ou     zz' 
ow  =  —  =  — 


ov 


az 


IMAGINARY  QUANTITIES.  21 


therefore  the  modulus  of  iv  is  equal  to  ^^^^j  and  the  direction- 
al 
coefficient  to  (—  cos  a  +  i  sin  a),  and  we  have 

w  =  ;2=:  (—  cos  a  +  i  sin  a). 
az 

In  the   special  case  when  az  is  perpendicular  to  ^,  a  =  90°, 

and  we  obtain 

.zz* 
w  =  I • 


Second :  In  what  relation  stand  two  groups,  of  three  points 
each,  z,  z',  z"  and  w,  w',  w",  if  between  them  the  equation 


z"  —  z     w'  —  w 
hold  (Fig.  6)  ?    We  have  immediately  the  proportion 

z'  —  z  :  z"  —  z  =  iv'  —  w:  w"  —  w, 

and  since  the  differences  denote  the  differences  of  correspond- 
ing points  in  length  and  direction, 
it  follows  directly  that  the  trian- 
gles z'zz"  and  lo'wv}"  are  directly 
similar. 

We  here  interrupt  these  con- 
siderations, passing  over  the  con- 
struction of  powers,  as  not  neces- 
sary for  our  purposes.  It  may, 
however,  be  noted  that  in  the  case  of  real  integral  exponents 
the  construction  follows  directly  from  the  repeated  appli- 
cation of  multiplication.^  One  other  remark  may  not  be  out 
of  place  here.  If  we  have  an  analytical  relation  between 
any  quantities  and  carry  out  the  analytical  operations  on  both 

1  For  powers  of  any  kind  we  refer  to  the  article :  "Ueber  die  geome- 
trische  Darstellung  der  Werthe  einer  Potenz  mit  complexer  Basis  und 
complexem  Exponenten."  (Schlomilch's  Zeitschrift  fur  Mathematik 
und  Physik,  Bd.  V.,  S.  345.) 


Fig.  6. 


22  THEORY  OF  FUNCTIONS. 

sides  of  the  equation  geometrically,  we  arrive  at  the  same 
point  by  two  different  methods  of  construction.  Hence,  every 
analytical  equation  contains  at  the  same  time  also  a  geometri- 
cal proposition.  Thus,  for  example,  it  may  readily  be  seen 
that  the  identity 

-— i —  =  v-\ — 


furnishes  the  proposition  that  the  diagonals  of  a  parallelogram 
bisect  each  other.^  By  means  of  a  geometrical  construction 
we  can  also,  among  other  things,  render  the  difference  between 
a  convergent  and  a  divergent  series  quite  evident.  As  is  well 
known,  the  geometrical  progression 

has  for  its  sum  the  value  ,  only  when  mod  2  <  1.     If  we 

1  —  z 

now  assume  an  arbitrary  point  z  and  construct  in  the  manner 
given  above  the  points  1,  1+z,  1  +  z  -{-  z^,  1  +  z  +  z^  -\-  z^,  etc., 
and  if  we  join  these  points  successively  by  straight  lines,  we 
obtain  a  broken  spiral.  If  then  the  point  z  be  so  situated  that 
mod  2  <  1,  i.e.,  oz  <  ol,  the  points  of  the  spiral  approach,  on 
windings  which  become  more  and  more  contracted,  that  point 

which  can  also  be  obtained  by  the  construction  of But 

■^  1-z 

if  mod  «  ^  1,  the  windings  of  the  spiral  become  steadily  wider, 
and  an  approximation  to  a  fixed  point  does  not  occur. 

3.  The  manner  of  representing  geometrically  complex  values 
by  points  in  a  plane  already  discussed,  also  gives  us  a  clear 
picture  of  a  complex  continuous  variable.  For,  if  we  imagine 
a  series  of  continuous,  successive  values  of  z  =  x-[- iy,  and 
therefore  also  a  series  of  continuous  successive  values  of  x  and 
y  (paired),  and  if  we  represent  each  value  of  z  by  a  point,  these 

1  "We  refer  those  who  wish  to  follow  out  still  further  the  line  of  thought 
connected  with  this  to  the  remarkable  article  by  Siebeck:  "Ueber  die 
graphische  Darstellung  imaginarer  Funktionen."  {Crelle's  Journ.,  Bd. 
55,  p.  221.) 


IMAGINARY  QUANTITIES.  23 

points  will  likewise  form  a  contiuuous  succession,  i.e.,  in  their 
totality  a  line.  Hence,  if  the  variable  z  change  continuously, 
the  point  representing  z  describes  a  continuous  line.  Since  in 
this  process  the  real  variables  x  and  y  can  each  vary  quite  inde- 
pendently of  the  other,  the  point  representing  z  can  also  describe 
an  arbitrary  line.  It  deserves  to  be  especially  mentioned  here 
that  for  the  continuity  of  the  variation  of  z  it  is  not  at  all 
necessary  for  the  line  described  by  the  corresponding  point 
to  be  a  curve  proceeding  according  to  one  and  the  same  mathe- 
matical law,  i.e.,  for  the  quite  arbitrary  relation  in  which  x  and 
y  must  stand  to  each  other  in  every  position  of  the  point  to 
be  always  expressible  by  the  same  equation  (or,  indeed,  by  any 
equation  whatever).  In  order  that  the  variation  of  z  may  be 
continuous,  it  is  necessary  only  for  the  line  to  form  a  continu- 
ous trace.  A  few  examples  may  ^ 
make  this  clear.  Suppose  the 
variable  z  begins  its  variation 
with  the  value  z  =  0,  and,  after 

passing    through     a    series     of     o  >ja~      xa 

values,  acquires  a  real  positive  fig.  t. 

value  a,  which  may  be  represented  by  the  point  a  (Fig.  7) 
on  the  a;-axis,  the  distance  oa  being  equal  to  a.  Now  the  vari- 
able z  (to  express  ourselves  more  briefly,  instead  of  saying,  the 
movable  point  which  represents  the  corresponding  value  of  the 
variable  z)  can  pass  from  o  to  a  on  very  different  paths.  Firstly, 
it  may  assume  between  o  and  a  only  real  values,  in  which  case 
y  remains  constantly  =  0  and  x  increases  from  0  to  a.  The 
variable  describes  the  straight  line  oa.  Secondly,  let  the  vari- 
able move  along  the  broken  line  oBCa  formed  of  three  sides  of 
a  rectangle  in  which  oB  =  b.  In  this  case  x  is  constantly  =  0 
from  o  to  B,  and  y  increases  from  0  to  b,  so  that  at  B,  z=ib; 
then  let  y  maintain  the  acquired  value  b,  and  x  increase  from  0 
to  a,  so  that  at  C,  z  assumes  the  value  a-\-ib;  finally,  from  C 
to  a,  let  X  remain  constantly  =  a  and  y  decrease  from  b  to  0. 
Thirdly,  the  variable  z  may  first  move  on  the  principal  axis 
from  o  to  ^  a,  and  then  run  along  a  semicircle  described  round 
the  point  |  a  as  centre  with  a  radius  \  a.     This  example  illus- 


24  THEORY  OF  FUNCTIONS. 

trates  at  tlie  same  time  the  transference  of  the  origin.  On 
account  of  the  circular  motion  round  the  point  f  a,  the  course 
of  the  real  variable  becomes  much  simpler,  if  we  put 

z  —  ^a  =  z'  =  r  (cos  <l>  +  i  sin  ^). 

The  radii  vectors  are  then  measured  from  the  point  |  a.  Now 
at  the  origin  z=0 ;  therefore  z'=  — f  a,  and  consequently  r=^a 
and  <f>  =  ir.  On  the  way  from  o  to  ^  a,  ^  remains  constantly 
=  TT,  and  r  decreases  from  f  a  to  ^  a,  so  that  at  the  beginning 
of  the  circle  z'  =  —  \a,  and  therefore  z  =  \a.  Now  in  describ- 
ing the  circle,  r  remains  constantly  =  \a  and  ^  decreases  from 
TT  to  0,  so  that  at  a,  z'=+\a,  and  therefore  z=a.  We  have 
here  assumed,  as  we  shall  always  do  in  the  future,  that 
the  angle  of  inclination  «/»  of  a  complex  quantity  increases 
from  the  direction  of  the  positive  ic-axis  toward  the  positive 
2/-axis,  and  we  shall  call  this  way  of  moving  the  direction  of 
increasing  angles.  From  these  examples  it  can  be  seen  that 
a  very  essential  difference  exists  between  a  variable  quantity, 
which  is  allowed  to  assume  only  real  values  and  one'  which 
may  assume  also  imaginary  values.  While  by  means  of  two 
definite  values  of  a  real  variable,  the  intermediate  series  of 
values,  which  the  variable  must  assume  in  order  to  pass  from 
the  first  to  the  second,  is  completely  determined,  this  is  by 
no  means  the  case  with  a  complex  variable ;  indeed,  there  are 
infinitely  many  series  of  continuous  values,  which  lead  from 
one  given  value  of  a  complex  variable  to  another  definite  value. 
Geometrically  expressed,  it  may  be  thus  stated :  a  real  variable 
can  proceed  only  by  a  single  path  from  one  point  to  another, 
namely,  on  the  intermediate  portion  of  the  principal  axis. 
On  the  contrary,  a  complex  variable,  even  when  the  initial  and 
final  values  are  real,  can  leave  the  principal  axis  and  pass  from 
the  one  point  to  the  other  on  an  infinite  number  of  lines  or 
paths.  If  the  initial  and  final  values,  one  or  both,  be  complex, 
the  same,  of  course,  holds ;  and  the  variable  can  take  arbitrary 
paths  in  passing  from  the  one  point  to  the  other. 


FUNCTIONS  OF  A   COMPLEX  VARIABLE.  25 

SECTION   11. 

FUNCTIONS    OF   A    COMPLEX   YARIABLE   IN   GENERAL. 

4.  In  passing  next  to  the  consideration  of  functions  of  a  com- 
plex variable,  we  begin  with  the  elementary  idea  of  a  function 
of  a  variable  quantity,  by  which  is  understood  any  expression 
formed  by  the  mathematical  operations  to  which  the  variable 
is  subject ;  but  we  shall  have  to  amplify  this  idea  later.  In 
former  times,  the  words  "  function  of  a  quantity "  signified 
merely  what  is  at  present  called  a  power.  It  is  only  since  the 
time  of  John  Bernoulli  that  this  term  has  been  applied  in  its 
extended  meaning,  signifying  not  only  the  raising  to  a  power, 
but  all  kinds  of  mathematical  operations,  or  any  combination 
of  the  latter.  In  more  recent  times,  however,  it  has  become 
necessary  to  enlarge  still  further  the  concept  of  a  function, 
and  to  dispense  with  the  necessity  of  the  existence  of  a  math- 
ematical expression  for  it.  For  if  one  variable  be  expressed 
in  terms  of  another,  so  that  the  former  is  a  function  of  the 
latter,  the  essential  feature  of  the  connection  between  the  two 
appears  in  the  fact  that  for  every  value  of  the  one  there  is  a 
corresponding  value  (or  several  corresponding  values)  of  the 
other.  Now  it  is  this  correspondence  of  the  values  of  the  func- 
tion on  the  one  hand,  and  of  the  independent  variable  on  the 
other,  which  we  especially  keep  in  view.  It  is  also  this  which 
is  made  prominent  wherever  we  recognize  the  dependence  of 
one  quantity  upon  another,  without  being  able  to  state  the  law 
of  this  dependence  in  the  form  of  a  mathematical  expression. 
To  take  a  familiar  example,  we  know  completely  the  depend- 
ence of  the  expansion  of  the  saturated  vapor  of  water  upon 
its  temperature  in  such  a  way  that,  after  the  observations  made 
and  tables  constructed  from  them,  we  can  determine,  within 
certain  limits,  the  expansion  of  the  vapor  for  every  value  of 
its  temperature.  But  we  do  not  possess  a  formula  derived 
from  theory,  by  means  of  which  we  could  calculate  the  expan- 
sion for  a  given  temperature.  Notwithstanding,  however,  the 
lack  of  such  a  mathematical  expression,  we  are  still  justified 


26  THEORY  OF  FUNCTIONS, 

in  considering  the  expansion  as  a  function  of  the  temperature, 
because  to  each  value  of  the  latter  a  definite  value  of  the 
former  appertains.  The  case  is  the  same  with  algebraic  func- 
tions in  the  general  sense,  i.e.,  with  functions  which  arise  by- 
connecting  one  variable  with  another  by  means  of  an  algebraic 
equation.  As  is  well  known,  equations  of  higher  degrees  can- 
not in  general  be  solved,  and  therefore  one  variable  cannot  be 
expressed  in  terms  of  the  other.  But  since  we  know  that  to 
every  value  of  the  latter  corresponds  a  definite  number  of  values 
of  the  former,  we  may  consider  the  former  as  a  function  of 
the  latter.  Besides,  functions,  whether  they  admit  of  being 
expressed  mathematically  or  not,  possess  some  characteristic 
properties,  usually  very  small  in  number,  by  which  they  can 
be  determined  completely,  or  at  least  except  as  to  a  con- 
stant factor  or  an  additive  constant.  Hence  we  can  replace 
the  expression  of  the  function  by  its  characteristic  properties. 
If  now  we  suppose  that,  within  a  certain  interval  of  the 
values  of  the  independent  variable,  a  function  is  determined 
only  by  giving  or  arbitrarily  assuming  the  value  of  the  latter 
which  corresponds  to  each  value  of  the  former,  yet  in  such 
a  way  that,  in  general,  to  continuous  changes  of  the  variable 
correspond  also  continuous  variations  of  the  function,  then  a 
distinction  occurs,  according  as  only  real  values  are  assigned 
to  the  variable  in  the  given  interval,  or  complex  values  are  also 
included  in  the  sphere  of  our  discussion.  In  the  former  case 
—  the  variable  assuming  only  real  values  —  we  can,  indeed, 
assume  quite  arbitrarily  the  values  of  the  function  which  are 
to  be  attached  to  those  of  the  variable,  and  let  the  one  set 
correspond  to  the  other  conformably  with  continuity.  In  this 
case  we  can  always  find  for  the  function  an  analytical  expres- 
sion which  shall  represent  its  values  within  the  interval  in 
question ;  for,  if  not  in  any  other  way,  this  is  always  possible 
by  means  of  the  series  which  proceed  according  to  the  sine  or 
cosine  of  the  multiples  of  an  arc.  As  is  well  known,  this  is 
possible  even  when  the  function  in  isolated  places  suffers  an 
interruption  of  its  continuity.  But  when  complex  values  enter 
into  the  discussion,  we  are  no  longer  at  liberty  to  choose  arbi- 


FUNCTIONS  OF  A   COMPLEX  VARIABLE.  27 

trarily  a  series  of  continuous  complex  values,  and  consider 
them  as  the  values  of  a  function  belonging  to  a  continuous 
series  of  values  of  a  complex  variable.  We  shall  consider  this 
point  more  fully  later.  In  the  meantime  we  wish  only  to  call 
attention  to  the  fact  that,  even  when  in  a  complex  variable 
w  =  u-{-  iv,  the  quantities  u  and  v  are  functions  of  the  real 
constituents  x  and  y  oi  the  variable  z  =  x  -\-  iy,  yet  w  on  that 
account  need  not  be  a  function  of  z.  We  shall  first  discuss  this 
condition  somewhat  more  fully  in  the  following  paragraph. 

5.'  Let  us  first  assume  that  we  have  under  discussion  an 
expression  representing  a  function  of  a  complex  variable 
z  =  x-\-iy ;  then  this  can  be  reduced  again  to  the  form  of  a 
complex  quantity,  i.e.,  to  the  form 

w  =  u  +  iv, 

wherein  u  and  v  denote  real  functions  of  x  and  y.  But  now 
every  expression  of  the  latter  form  is  not,  conversely,  at  the 
same  time  also  a  function  of  z ;  for,  that  this  may  be  so,  it  is 
necessary  for  the  real  variables  x  and  y  to  occur  in  u  +  iv, 
only  in  the  definite  combination  x  +  iy-  It  is  evident  that 
we  can  easily  form  functions  of  x  and  y  in  which  this  is  not 
the  case,  as,  for  instance,  x  —  iy,  a^ -{-y%  2x  +  iy.  These 
are,  it  is  true,  functions  of  x  and  y,  but  not  of  a;  +  iy ;  they 
are  complex  functions,  but  not  functions  of  a  complex  variable, 
—  concepts,  which  must  therefore  be  well  distinguished.  Thus 
the  problem  arises  to  inquire  what  conditions  must  be  satisfied 
by  a  given  expression  w  =  u  -\-  iv,  in  which  u  and  v  signify  real 
functions  of  x  and  y,  in  order  that  the  expression  may  be  a 
function  of  z  =  x  -\-  iy.  To  find  these  conditions,  we  differ- 
entiate w  partially  as  to  x  and  y ;  then,  if  w  shall  in  the  first 
place  be  a  function  of  z,  we  have 


8io 

_  dw  Sz 

8x~ 

'  dz  8x 

8io 

_  die  8z 

%^ 

~dz8y' 

28  THEORY  OF  FUNCTIONS. 

or,  since  8^  =  ^'  8^  =  ^' 

the  following  ^  =  ;^'8^='^- 

Hence  we  obtain,  as  tlie  necessary  condition  that  w  shall  be  a 
function  of  z,  the  equation 

8w  _  .8w 

Sy~^8x' 

Conversely,  it  can  easily  be  proved  that  this  condition  is  suffi- 
cient, i.e.,  that  a  function  w  of  a;  and  y,  which  satisfies  this 
equation,  will  always  be  a  function  of  z.  For,  if  in  the  com- 
plete differential 

dw  =  —  dx-]-  —  dy, 
6x  Sy 

we  substitute  i  —  for  — , 
8a;  8y 

we  obtain  dw  =  -^  (dx  +  idy)  =  —  dz. 

8x^  ^^      8a; 

If,  however,  by  means  of  z  =  x  +  iy,  the  variable  x  be  elimi- 
nated from  the  function  w  before  differentiation,  and  if  the 
partial  derivatives  as  to  y  and  z,  derived  after  the  elimination, 
be  distinguished  from  the  former  by  parentheses,  we  have 

by  subtracting  this  expression  for  dw  from  the  former,  we  get 

But  since  dy  and  dz  are  entirely  independent  of  each  other, 
separately  must  ■ 

8w\    Q      /8w\^8w 
,%/       '     \8zJ      Sx' 

From  the  first  of  these  equations  it  follows  that  w,  after  the 
elimination  of  x,  no  longer  contains  y,  but  is  a  function  of  z 


FUNCTIONS  OF  A   COMPLEX  VARIABLE.  29 

only.     Then  ( -—  )  and  —  have  the  same  meaning,  and  there- 
\oz  J  oz 

fore   the  second  equation  gives  —  =  — ,  the  same  result  as 

dz      ox 

before.     Therefore  the  above  relation 

^  '  8y~    ^ 

is  the  necessary  and  sufl&cient  condition  to  ensure  the  stated 
functionality  of  iv.  From  this  also  follow  the  equations  of 
condition  for  the  real  parts  u  and  v.  It  u-\-  iv  be  substituted 
for  IV,  we  obtain 


8ti  , 

8y 

.8v 

'■(1- 

.8v\ 
'  8x/ 

and  then 

by  equating  real  and 

imaginary  parts, 

(2) 

8u 
8^~ 

St) 

'-W 

8u 

8v 
8x 

Finally,  we  can  establish  for  each  of  these  functions  a  single 
equation  of  condition.  For,  differentiating  each  of  the  above 
equations  partially  as  to  x  and  y,  and  eliminating  v  and  u  in 
turn,  we  obtain 

/o\  S^u  ,   8ht     p,       -,  8^v  ,  8^v     p, 

so  that  neither  of  the  functions  u  and  v  is  arbitrary,  but  each 
one  must  satisfy  the  same  partial  differential  equation.  As 
is  well  known,  partial  differential  equations  do  not  characterize 
particular  functions  but  general  classes  of  functions.  Thus  the 
function  w  of  the  complex  variable  z  =  x  +  iy  is  given  by  equar 
tion  (1),  and  the  real  constituent  parts  of  such  a  fimction  by 
(2)  and  (3). 

6.  If  we  still  hold  to  the  supposition  that  the  function  w  is 
given  by  an  expression,  an  important  inference  can  be  drawn 
from  equation  (2).     To  the  increment  dz  of  z  corresponds  the 


30  THEORY  OF  FUNCTIONS. 

increment  —  dz  of  w.    By  introducing  the  quantities  u,  v  and 
dz  ■, 

X,  V  into  the  derived  fimction  —  we  obtain 
'^  dz 


dw     du  +  idv 


l^+l^^^+ll:^^'^) 


dz      dx  +  idy  dx  +  idy 

But  now,  when  the  variable  z  is  represented  by  a  point  in 
the  a;i/-plane,  this  point  can  move  in  any  arbitrary  direction, 
and  the  differential  dz  =  dx+  idy 

represents  the  infinitely  small  straight  line  which  indicates 
the  change  of  place  of  z  in  magnitude  and  direction.  This 
infinitely  small  straight  line  can  therefore  be  drawn  from  z  in 
any  arbitrary  direction.  Now,  however,  the  preceding  expres- 
sion for  —  shows  that  it  is  not  independent  of  dz,  but  changes 
dz 

its  value  with  the  direction  of  dz.     To  make  it  still  clearer,  let 

us  introduce  the  differential  coefficient  -^,  which  indicates  the 

dx 

direction  of  dz,  into  the  expression  for  — .  Dividing  numera- 
tor and  denominator  by  dx,  we  obtain 

Su      8udy      .f8v      8v  dy'\ 
dw_h;'^8ydx^^\Ex'^8ydx)^  (*) 

^  1  +  i^  ' 

dx 

from  which  it  follows  that  — ,  in  fact,  changes  its  value  with 
,  dz 

that  of  -^,  when  no  relation  exists  between  the  four  differential 
dx 

coefficients  — ,  -^,  — ,  — .     But  if  we  take  the  equations  (2) 
6x   oy    8x   oy  ^ 

into  consideration  and  by  means  of  them  eliminate,  say,  — 
and  -— ,  we  obtain 

dw      \8x       6xJ\         dxj      8u  ,    .hv 

dz  1  _i_  j^  S-^'       S^' 

dx 


FUNCTIONS  OF  A   COMPLEX  VARIABLE.  31 

thus  —  becomes  independent  of  -^  and  hence  also  of  dz.  If 
therefore  w  be  a  function  of  the  complex  variable  z  =  x-\-iy,  the 

derivative  —  is  independent  of  dz  and  Juis  the  same  value  in 
dz 

ivhatever  direction  the  infinitely  small  movement  may  take  place. 
If  we  call  the  different  paths  which  the  variable  may  take 
the  modes  of  variation,  we  can  say  that  the  derivative  is  inde- 
pendent of  the  mode  of  variation  of  the  variable  z.  In  the 
case  of  a  function  of  a  real  variable,  the  change  of  the  variable 
itself  does  not  make  any  essential  difference,  because  this 
change  can  only  consist  of  an  increase  or  decrease  of  the 
variable.  In  the  case  of  functions  of  a  complex  variable,  how- 
ever, the  different  ways  in  which  the  variable  can  change  play 
an  important  part,  and  hence  the  proposition  just  established, 
that  the  derivative  of  a  function  of  a  complex  variable  is 
independent  of  the  mode  of  variation  of   the  variable,  is  of 

great   importance.     And  it  is   only  when  —   is  completely 

independent,  i.e.,  both  of  the  length  and  of  the  direction  of  this 
infinitely  small  straight  line,  that  the  idea  of  the  derived 
function  becomes  as  definite  as  it  is  in  the  case  of  real 
variables. 

Until  now  we  have  been  assuming  that  the  function  w  is 
given  by  a  mathematical  expression  in  terms  of  z.  If  we  now 
give  up  this  assumption,  we  must,  in  order  that  the  derivative 
of  the  function  lo  may  have  a  definite  meaning,  still  add  the 
requirement  that  it  be  independent  of  the  differential  dz. 

The  fulfilment  of  this  requirement,  however,  is  sufficient  to 
characterize  to  as  a  function  of  ic  +  iy,  for  from  it  follow  again 
our  former  conditions  (1),  (2)  and  (3).     If  the  expression  (4)  for 

—  is  to  be  independent  of  dz,  or  what  is  the  same  thing,  of  -/. 
dz  ^  «fcc 

the  equation  resulting  from  it, 


dw 
dz 


hi _  .8v      /' .dw _8u_  . 8v\ dy _r. 
~hx~^hx\dz      hj     ^8yjdx~    ' 


32  THEORY  OF  FUNCTIONS. 

must  be  satisfied  for  every  value  of  -^'     Therefore,  we  obtain 

dx 

dw  _8u      .  8v  _  8iv 
dz      Bx       8x     8x 

.dw_Bu,   .Sy_8w 

,  8w      .8w 

or,  as  above,  i;~  =  *  i; — 

Sy        8x 

In  accordance  with  this,  Eiemann  ^  has  defined  a  function  of 
a  complex  quantity  in  the  following  way :  "  A  variable  com- 
plex quantity  w  is  called  a  function  of  another  variable  complex 
quantity  z,  if  it  so  change  with  the  latter  that  the  value  of  the 

derivative  —  is  independent  of  the  value  of  the  differential  dz." 

Or,  as  it  is  expressed  in  another  place  ^ :   ''  If  w  change  with 

x-\-iy  in  conformity  to  the  equation  —  =  i  — •" 

8y        8x 
It  can  also  be  easily  proved  that,  if  w  be  a  function  of  z,  the 

derivative  —  must  likewise  be  a  function  of  z.     For.  from  the 
dz 

equations 


follow 
and 


consequently 


dtv  _  8iv  _ 
'dz~Jx~ 

_18w 
~  i  8y 

8  fdiv\  _ 
8x\dz) 

1  8'w 
i  8xSy 

8_fdw\  _ 
8y\dz) 

8'w 
'8x8y' 

8  fdw\ 
8y\dzJ 

-  i  ^  (^^\ 
8x\dz/ 

dw 


and  therefore  ■—  also  satisfies  equation  (1). 

1  Gfrvndlagen  fur  eine  allgemeine  Theorie  der  Funktionen  einer  verdn- 
derlichen  complexen  Chrosse.     S.  2. 

2  "Allgemeine  Voraussetzungen,"  etc.,  Crelle's  Journ.,  Bd.  54,  S.  101. 


FUNCTIONS  OF  A    COMPLEX   VARIABLE.  33 

Further,  if  w  be  a  function  of  z  =  x-\-iy,  and  if  z  be  a  func- 
tion of  ^  =  ^  4-  irj,  then  w  is  also  a  function  of  ^.  For,  as 
above,  p.  28, 

ox  Sx 

and  similarly         dz=—(di  +  idrj), 

consequently         dw  =  —  —  (d^^  idrj) ; 

thus  the  partial  differential  coefficients  of  w  as  to  |  and  17  are 

8w  _  8iv  Sz     8w  _  .  8w  8z 

hence  ^  =  ^•^. 

and  therefore  w  is  also  a  function  of  ^  +  irj. 

7.   The  condition  just  established  possesses  a  definite  geo- 
metrical meaning,  which  remains  to  be  discussed. 
If,  as  above, 

z  =  x-\-iy  and  w  =  u  +  iv, 

X  and  y  are  the  rectangular  co-ordinates  of  a  point  in  the 
2-plane,  and  u  and  v  the  rectangular  co-ordinates  of  a  point  w  in 
the  same  or  in  another  plane.  If,  now,  w  be  a  function  of  z, 
the  position  of  the  point  w  depends  upon  the  position  of  the 
point  z,  and  if  z  describe  a  curve,  w 
describes  a  curve  depending  upon 
the  latter ;  in  short,  if  iv  be  a  defi- 
nite function  of  z,  the  entire  system 
consisting  of  the  points  w  is  in  a 
definite  dependence  upon  the  sys- 
tem formed  by  the  points  z.  Rie- 
mann  calls  then  the  system  of  the  ^^^-  ^^ 

points  w  the  conformed  representation  of  the   system  of  the 
points  z.     In  accordance  with  the  above  condition,  the  two 


84  THEORY  OF  FUNCTIONS. 

figure-systems  stand  in  a  quite  definite  relation  to  each  other, 
which  always  holds  when  w  is  a  function  of  z. 

Let  z'  and  z"  (Fig.  6)  be  two  points  infinitely  near  to  a  third 
point  z,  and  let  the  infinitely  small  strokes  joining  them  and 
running  in  different  directions  be  ' 

zz'  =  dz',  zz"  =  dz". 

Further,  let  the  points  which  correspond  to  the  points  2,  z',  z" 
be  w,  w',  w",  and  let  the  infinitely  small  strokes  joining  the 
latter  be 

ww'  =  dw',  ww"  =  dw"  \ 

If,  now,  —  is  to  have  the  same  value  for  every  direction  of  dz, 

CLZ 

then  dw' _dw"         dw'  _dz' 

'dz'~"dz"'  °^  dw^'~dz^'' 

But  the  differentials  can  now  be  replaced  by  the  differences  of 
the  infinitely  near  points ;  that  is, 

dz'  =  z'  —  z,  dw'  =^w'—Wf 

dz"  —  z"  —  z,   dw"  —  w"  — 10, 
and  we  have 

w'  —  w  _z'  —z^ 
w"  —  w     z"  —  z^ 

therefore,  by  §  2,  the  triangles  z'zz"  and  w'ww"  are  similar  to 
each  other;  i.e.,  the  angles  z'zz"  and  w'ww"  are  equal  to  each 
other,  and  the  included  sides  are  proportional.  But  since  this 
must  hold  for  any  pair  of  corresponding  points  z  and  w,  the 
figure  described  by  the  point  w  is  in  its  infinitesimal  elements 
similar  to  that  described  by  the  point  z,  and  two  intersecting 
curves  in  the  w-plane  form  with  each  other  the  same  angle  as 
that  formed  by  the  corresponding  curves  in  the  z-plane.     In 

this  connection  it  must  be  noticed  that  —  is  supposed  to  be 

1  We  note  that,  even  if  —  be  independent  of  dz,  yet  dw,  which  =  —  dz, 
dz  dz 

in  general  changes  its  direction  and  magnitude  with  dz. 


FUNCTIONS  OF  A   COMPLEX  VARIABLE.  35 

neither  zero  nor  infinite.  We  shall  see  later  that  these  cases 
are  exceptions.^  Siebeck  terms  the  dependence  of  the  system 
10  upon  the  system  z  conformation;  and  on  account  of  the 
property  that  any  two  pairs  of  corresponding  curves  include 
equal  angles,  isogonal  conformation}  The  simplest  isogonal 
conformations  are  similarity  and  circular  conformation^  (intro- 
duced into  geometry  by  Mobius).     In  the  former,  w  =  az  +  h; 

in  the  latter,  w  =  — ^^,  wherein  a,  b,  c,  d  denote  constants. 
cz  +  d 

Collinearity  and  affinity  are  not  isogonal  conformations ;  these 
do  not  admit  of  being  represented  by  functional  relations 
between  two  complex  variables. 

The  simple  function  w  =  z^ 

may  serve  as  an  example. 

We  obtain  here        w  =  o?  —  y^  -\- 2  ixy, 
and  hence  u  =  x^  —  y"^,       v  =  2xy, 

^=2x,  |^=2y, 

Sx  '  8x        ^' 

^  =  -2y,         ^^2x, 

which  verify  the  equations  of  condition  (2).  Let  now  z  de- 
scribe, for  instance,  the  ?/-axis,  so  that  x=  0,  then  z  =  iy  and 
w  =  —  'if;  hence  w  describes  the  negative  part  of  the  principal 
axis  and  only  this,  so  that,  when  z  goes  from  a  through  o  to 
b,  w  moves  from  a'  to  o,  and  then  back  again  to  6';  a'  and  b' 
coincide  when  ao  is  assumed  equal  to  ob  (Fig.  8).  Let  z  fur- 
ther describe  a  circle  with  radius  r  round  the  origin,  so  that 
when 

z  =  r(cos  <^  4-  *  sin  <}>), 

r  remains  constant ;  then 

w  =  r^(cos  2  <^  -f  I  sin  2  </>), 

1  Cf .  §  40. 

2  Known  also  as  isogonal  or  orthomorphic  transformation.     (Tr.) 
8  Called  also  bilinear  or  homographic  transformation.     (Tr.) 


36 


THEOEY  OF  FUNCTIONS, 


and  w  also  describes  a  circle  round  the  origin  with  radius  r^. 
But  since  the  angle  2  <j>  oi  the  ?<;-plane  corresponds  to  the 
angle  <f>  of  the  2;-plane,  lo  describes  its  circle  twice  as  rapidly 
as  z.  For  instance,  if  z  describe  a  semicircle  from  a  to  b  in 
the  direction  of  increasing  angles,  w  describes  a  complete 
circle  from  a'  to  the  point  b'  (which  coincides  with  a').  But 
the  angles,  which  the  straight  line  and  the  circle  form  with 


Fig.  8. 


each  other  in  z  and  in  w,  are  both  right  angles.  If  we  let  z 
describe  a  straight  line  cd  passing  through  the  point  1  and 
parallel  to  the  ?/-axis,  w  will  describe  a  parabola.  This  result 
can  be  easily  obtained,  —  since  in  this  case  x  is  constant  and 
equal  to  1,  —  by  substituting  the  value  of  x  in  the  equations 
u  —  o^  —  'if'  and  v  =  2xy  and  eliminating  y ;  thereby  we  obtain 
the  equation  v^  =  4(1  —  u)  between  the  co-ordinates  u  and  v 
of  the  point  w,  which  shows  that  the  locus  of  w  is  a  parabola, 
with  its  vertex  at  1,  its  focus  at  o,  and  of  which  the  param- 
eter, the  ordinate  at  the  focus,  is  2.  By  examining  the 
tangents  at  the  points  of  intersection  c'  and  d',  which  cor- 
respond to  c  and  d,  it  is  easily  verified  that  the  parabola 
cuts  the  circle  in  the  w-plane  under  the  same  angle  as  the 
straight  line  cd  cuts  the  circle  in  the  2;-plane.  But  finally, 
in  order  to  illustrate  an  exception  by  an  example,  let  z  de- 
scribe the  principal  axis ;  then  z  remains  real,  hence  w  is  posi- 
tive and  therefore  describes  the  positive  part  of  the  principal 
axis.      But  the  latter  forms  with  the  negative  part  of  the 


MULTIFORM  FUNCTIONS.  37 

principal  axis,  which  corresponds  to  the  ^/-axis  in  the  z-plane, 
an  angle  of  180°,  while  the  x-  and  y-axis  in  the  z-plane  form  an 
angle  of  90°.  Therefore,  in  the  vicinity  of  the  origin  the 
similarity   of  infinitesimal   elements  does  not  occur,  and,  in 

fact,  at  this  point  the  derivative  —  =  2z  becomes  zero. 

dz 


SECTION  III. 

MULTIFORM   FUNCTIONS. 


8.  The  introduction  of  complex  variables  also  throws  a 
clear  light  on  the  nature  of  multiform  (many-valued)  functions. 
For,  since  a  complex  variable  may  describe  very  different 
paths  in  passing  from  an  initial  point  Zq  to  another  point  Zj, 
the  question  naturally  suggests  itself,  whether  the  path  de- 
scribed cannot  affect  the  value  w ,  which  a  function,  starting 
with  a  definite  value  Wq  corresponding  to  Zq,  acquires  at  the 
terminal  point  z^;  we  have  to  inquire  whether  the  curves 
described  by  w,  starting  from  Wq,  which  correspond  to  those 
described  between  Zq  and  Zj,  must  always  end  in  the  same 
point  Wi,  or  whether  they  cannot  also  end  in  different  points. 
Now,  in  the  first  place,  it  is  clear  that,  in  the  case  of  uniform 
(one-valued)  functions,  the  final  value  w^  must  be  independent 
of  the  path  taken ;  for,  otherwise,  the  function  would  be  capar 
ble  of  assuming  several  values  for  one  and  the  same  value  of 
z,  which  is  not  possible  with  uniform  functions.  This  reason, 
however,  does  not  apply  in  the  case  of  multiform  functions. 
Such  a  function  has,  in  fact,  several  values  for  the  same  value 
of  z,  and  hence  the  possibility,  that  different  paths  may  also 
lead  to  different  points  or  to  different  values  of  the  function, 
is  not  excluded  at  the  outset.  Let  the  variable  z  in  w  =  a/z, 
for  instance,  pass  from  1  to  4  by  different  paths,  and  let  the 
function  w  start  with  w  =  -{-l  corresponding  to  z  =  1 ;  then  it 
is  possible  that  some  of  the  paths  shall  lead  from  w  =  -}- 1  to 


38  THEORY  OF  FUNCTIONS. 

w  =  -\-2,  and  others,  on  the  contrary,  from  w  =  +  ltowj  =  —  2. 
This  is,  indeed,  actually  the  case  in  this  example.  For  let 
z  =  r(cos  <f)  +  i  sin  <^),  then  w  =  V^(cos  ^  </>-}-  i  sin  i  </>),  in  which 
by  Vr,  since  it  is  the  modulus  of  w,  is  to  be  understood  the 
positive  value  of  the  square  root  of  r.  Since  w  is  to  start 
with  the  value  + 1,  the  initial  values  of  the  real  variables  are 
r  =  1  and  «^  =  0.  If  z  describe  a  path  between  1  and  4,  which 
does  not  enclose  the  origin,  —  for  instance,  the  straight  line 
from  1  to  4,  —  then  cf>  arrives  at  the  point  4  with  the  value 
zero,  while  r  acquires  here  the  value  4 ;  hence  along  such  a  path 
w  receives  the  value  +  2.  If,  on  the  other  hand,  the  path  de- 
scribed by  z  between  1  and  4  go  once  round  the  origin,  then 
at  the  point  4,  «^  acquires  the  value  2  tt,  and  ^  <f>  the  value  tt, 
while  again  r  =  4 ;  hence  w  acquires  in  this  case  the  value 
-2.     (Cf.  also  §  10.)' 

Here  we  must  first  of  all  direct  our  attention  to  those  points, 
at  which  two  or  more  values  of  the  function  w,  in  general 
different,  become  equal  to  one  another.  Such  a  point  is,  for 
instance,  z  =  0  for  w  =  Vz ;  at  this  point  the  values  of  w,  in 
general  of  different  signs,  become  equal  to  zero. 

Let  us  next  consider  the  function  defined  by  the  cubic 
equation 

w^  —  w  +  z  =  0. 

If,  for  brevity,  we  put 


and  the  two  imaginary  cube  roots  of  unity 

-  1  +  W3  -  1  -  iV3       2 

Cardan's    formula    gives   for   the  three   roots   of   the  above 

1  We  have  in  view  in  these  considerations  the  (irrational)  algebraic 
functions,  and  hence  always  assume  that  the  number  of  values  which 
the  function  can  assume  for  the  same  value  of  the  variable  z  is  finite. 


MULTIFORM    FUNCTIONS.  39 

equation,  which  may  be  denoted  by  Wi,  w^,  w^,  the  following 
expressions : 

W2=ap-\-  o?q, 

Wg  =  «-^  +  aq. 

For  each  value  of  z,  w  has  in  general  the  three  values 
Wi,  W.2,  w^.  But  the  last  two  of  these  become  equal  when 
p  =  q,  which  occurs  when 

2 

z  = '• 


^27 
At  this  point  we  have 

If  now,  in  further  discussion  of  this  example,  we  assume 
that  the  variable  z  changes  continuously,  or  that  the  point 
representing  it  describes  a  line,  then  each  of  the  three  quan- 
tities, ^(,'l,  Wo,  tCs,  likewise  changes  continuously,  or  the  three 
corresponding  points  describe  three  separate  paths.     But  when 

2 

2  passes  through  the  point  z  = ,  both  functions,  Wg  and  Wg, 

V27 
assume  the  value  V^;  hence  the  two  lines  described  by  Wj 
and  iVg  meet  in  the  point  V^.  At  the  passage  through  this 
point  therefore  iV2  can  go  over  into  iv^,  and  w^  into  w,,  without 
interruption  of  continuity ;  indeed,  it  remains  entirely  arbitrary 
on  which  of  the  two  lines  each  of  the  quantities  W2  and  Wg  shall 
continue  its  course.  In  this  place  a  branching,  as  it  were,  of 
the  lines  described  by  the  quantities  w^  and  Wg  takes  place; 
hence  Riemann  has  called  those  j^oints  of  the  z-j)lane,  at  which 
one  value  of  the  function  can  change  into  another,  branchpoints. 

2 
In   our   example   therefore  z  =  — —    is   a  branch-point   (not 

w  =  y^Y     Figures  A  and   B   are  added  in  explanation.     In 

Fig.  A  the  three  lines  w^,  w^,  Wg  are  drawni  for  the  case  when 

z  describes  a  straight  line  parallel  to  the  ^/-axis  and  passing 

2 
through  the  branch-point  e  =  — m  (Fig.  B).     Therein,  however. 


40 


THEORY  OF  FUNCTIONS. 


the  line  Wi  is  represented,  for  clearness,  on  twice  as  large  a 
scale  as  the  remaining  lines  and,  to  save  space,  it  is  drawn 
nearer  to  the  ordinate  axis  than  it  really  runs.  The  w-points 
which  correspond  to  the  2;-points  are  denoted  by  the  same 
letters  with  attached  subscripts  1,  2,  3.     The  picture  of  the 


Fig.  a. 


Fig.  B. 


branching  is  rendered  still  clearer  by  following  the  path  of 
only  one  of  the  quantities,  say  Wg.  This  describes  the  line 
ftgCgrfg,  aud  approachcs  the  point  eg  =  62  =  V|^,  as  z  approaches 


the  point  e  = 


^ 


along  the  line  hcd\    should  z  now  pass 


through  this  point,  w^  could  continue  its  course  from 

62  =  63 = V^ 


MULTIFORM    FUNCTIONS.  41 

on  either  of  two  paths,  namely,  esfy^h^  or  e^f-^^^f  of  which  one 
as  well  as  the  other  can  be  considered  the  path  corresponding 
to  the  continuation  efgli  of  z\  the  way  open  to  Wg  is  in  fact 
divided  at  e,  =  63  into  two  branches.  When  z  goes  from  &  to  ^ 
through  the  branch-point  e,  then  Wg,  starting  from  ftg,  can  arrive 
at  lu  just  as  well  as  at  /ig;  and  the  same  is  true  of  w^  starting 
from  62-  III  case  the  path  of  z  leads  through  the  branch-point, 
the  final  value  of  the  function  remains  therefore  undeter- 
mined. If,  on  the  contrary,  z  describe  a  path  from  6  to  Ji, 
which  does  not  pass  through  a  branch-point,  the  final  value  of 
the  function  may,  it  is  true,  differ  according  to  the  nature  of 
the  path,  but  it  is  for  each  definite  path  of  z  always  completely 
determined.  The  figures  A  and  B  illustrate  this  also.  If  z 
move  from  h  through  d,  and  next  along  the  broken  line  through 
mtof  and  h,  then  iv^  moves  from  &g  through  dg,  and  next  along 
the  broken  line  through  mg  to  /g  and  h^;  w^  moves  from  62 
through  dg,  m2,  /2  to  h^ ;  Wg  acquires  then  the  definite  value  ^g, 
and  ?i'2  the  definite  value  /ia-  These  final  values  will  be  differ- 
ent, but  again  definite,  when  z  goes  round  the  branch-point  e 
on  the  other  side  along  the  dotted  line  through  p.  In  this  case 
Wg  goes  from  \  through  dg,  and  then  along  the  dotted  line 
through  p.^  to  /a  and  h^ ;  and  w«  goes  through  d2,  Pz^  fa  to  ^3.  In 
this  case  the  successive  values  of  the  function,  and  therefore 
also  the  final  values,  are  different  from  the  former,  but  again 
they  are  completely  determined. 

As  a  general  rule,  only  those  points  of  the  2;-plane,  at  which 
several  values  of  the  function  (elsewhere  unequal)  become 
equal,  are  also  branch-points.  An  exception  to  this  is  to  be 
mentioned  immediately. 

A  similar  branching  of  the  function  takes  place  at  those 
points,  at  which  w  becomes  infinite  and  therefore  discontinuous. 
Thus,  for  instance,  the  point  2;  =  0  is  a  branch-point  both  for 

the  function  w  =  — =  and  for  w  =  V^.     Further,  in  the  function 

■y/z 

determined  by  the  equation 

(z  —  b)(w  —  cy  —  z  —  a  or  w  =  c  +-y -, 


42  THEORY  OF  FUNCTIONS. 

in  which  a,  b,  c  denote  three  complex  constants,  and  therefore 
three  points,  z  =  a  is  a  branch-point,  at  which  all  three  values 
of  the  function  become  equal  to  «;  =  c.  Moreover,  at  2;  =  5  all 
three  values  of  w  become  infinite.  The  three  functions  suffer 
here  an  interruption  of  continuity,  and  hence  it  can  remain 
undetermined  on  which  path  each  is  to  continue  its  course, 
because,  when  the  function  makes  a  spring,  it  can  just  as  well 
spring  over  to  the  one  as  to  the  other  continuation  of  its  path. 
Therefore  z  =  &  is  likewise  a  branch-point.  Also,  as  a  general 
rule,  those  points  at  which  w  becomes  infinite  or  discontinuous 
are  branch-points.  Exceptions  to  this,  however,  may  occur; 
there  are  cases  in  which  points  are  not  branch-points,  although 
at  them  the  values  of  the  functions  are  either  equal  or  infinite. 
This,  for  the  present,  can  only  be  illustrated  by  examples.  In 
the  functions 

VI  —  z^  and      .         ■> 
Vl—  z" 

2  =  + 1  and  2  =  —  1  are  branch-points ;  on  the  contrary,  in 

^  1 

(z  —  a)  V2  and  ~ p' 

iz  —  a)yz 

2  =  a  is  not  a  branch-point,  although  the  values  of  the  func- 
tions at  this  place  are  in  the  first  case  both  zero,  and  in  the 
second  case  both  infinite.  For,  when  2  passes  through  the 
point  a,  then  2  —  a  as  well  as  V2  has  a  perfectly  definite,  con- 
tinuous progress  \  z  —  a,  because  it  is  uniform,  and  V2,  because 
-f  Va  cannot,  without  interruption  of  continuity,  suddenly  pass 
over  into  —  Va.  Hence  the  rational  functions  of  these  quan- 
tities have  at  this  point  a  definite  continuation  for  every  path 
described  by  2,  and  there  is  no  branching.  Accordingly,  the 
branch-points  are  to  be  looked  for  only  among  those  points 
at  which  an  interruption  of  continuity  occurs,  or  at  which 
several  values  of  the  function  become  equal ;  but  whether 
such  points  are  actually  branch-points  must  still  be  expressly 
determined. 


MULTIFORM   FUNCTIONS.  48 

9.   The  preceding  considerations  have  shown  that,  when  the 
variable  z  starting  from  an  arbitrary  point  Zq  describes  a  path 
to  another  point  z^,  which  leads  through  a  branch- 
point of  a  function  w,  the  latter  acquires  differ- 
ent values  at  Zi  according  as  it  is  allowed  to 
proceed    on   one  or  another  of    its    branches. 
Therefore,  in  the  case  of  such  a  path  of  z,  the 
value  of  IV  at  Zj  is  undetermined.      If,  on  the 
contrary,  z  describe  any  other  path,  not  leading 
through  a  branch-point,  w  acquires  at  Zi  a  defi- 
nite value,  and  it  will  now  be  shown  that  two 
paths,  both  of  which  lead  from  z^  to  Zj,  assign 
different  values  to  iv  at  z^,  only  tvhen  they  enclose 
a  branch-point.     To  that  end  we   first   prove   the   following 
proposition : 

Let  the  variable  z  in  jyassiiig  from  Zq  to  z^  describe  two  infinitely 
near  paths,  z^mzi  and  z^tiZi  (Fig.  9),  which  in  no  place  approach 
infinitely  near  a  point  at  which  either  the  function  w  becomes 
discontinuous  or  several  values  of  the  function  become  equal,  then 
the  function  w,  starting  from  Zq  with  one  and  the  same  value, 
acquires  at  Zj  the  same  value  on  both  paths. 

To  prove  this  proposition,  we  first  remark  that  the  different 
values  which  a  multiform  function  has  at  one  and  the  same 
point  z  can  differ  by  an  infinitely  small  quantity,  only  when 
the  point  z  lies  infinitely  near  a  point  at  which  several  values 
of  the  function  become  equal.  (Cf.  Figs.  A  and  B,  p.  40.  In 
that  example  the  lines  described  by  the  values  of  the  function 
approach  each  other  only  at  the  point  e,  while  at  all  other 
points  z  they  are  a  finite  distance  apart.)  Since  now,  according 
to  the  hypothesis,  the  two  paths,  Zomzi  and  Zonzi,  nowhere  ap- 
proach such  a  point,  the  different  values  which  w  can  have  at 
any  point  of  the  two  paths  differ  by  a  finite  quantity.  There- 
fore the  values  which  the  function  w  acquires  at  Zi  on  the  two 
paths,  Zomzi  and  Zgnzi,  must  either  be  equal  to  each  other  or 
differ  by  a  finite  quantity.  But  the  latter  alternative  cannot 
occur.  For,  if  we  suppose  that  two  movable  z-points  describe 
the  two  infinitely  near  paths,  Zomzi  and  z^nzj,  in  such  a  way 


44  THEORY  OF  FUNCTIONS. 

that  they  remain  always  infinitely  near  each  other,  and  if  we 
denote  the  value  of  the  function  along  the  one  line  by  w^,  and 
that  along  the  other  by  w„,  then  w^  and  w,^  along  both  lines  can 
differ  only  by  an  infinitely  small  quantity,  because,  by  the 
hypothesis  w  starts  from  Zq  with  the  same  value  on  both  paths, 
and  on  both  changes  continuously,  and  because,  further,  in 
passing  from  a  point  of  one  line  to  an  infinitely  near  point  of 
the  other,  the  continuity  is  not  broken.  Now,  if  w^  and 
w^  differed  by  a  finite  quantity  at  z^,  at  least  one  of  these 
functions  would  have  to  make  a  spring  in  some  place,  which 
is  excluded  by  the  hypothesis  that  the  two  paths,  Zomz^  and 
ZqUZ^,  shall  approach  no  point  at  which  an  interruption  of  con- 
tinuity occurs.  Consequently  w^  and  w„  cannot  differ  from 
each  other  by  a  finite  quantity,  and  hence,  according  to  the 
above,  they  are  equal  to  each  other. 

This  having  been  established,  if  we  now  suppose  a  series 
of  successive  paths  lying  infinitely  near  to  each  other,  all 
between  the  points  Zq  and  z^,  and  so  constructed  that  no  one 
of  them  approaches  a  point  at  which  either  discontinuity 
occurs  or  function-values  become  equal,  then  the  function 
acquires  on  all  these  paths  the  same  value  at  z^.  From  this 
folloM^s  the  proposition :  If  a  path  between  two  points,  z^  and  Zi, 
can  be  so  deformed  into  another  path  by  gradual  changes,  that 
thereby  no  one  of  the  above  defined  critical  points  is  passed  over, 
then  the  function  acquires  at  Zj  the  same  value  on  the  second  path 
as  on  the  first.  This  conclusion  holds  also  in  the  case  when 
two  points,  Zo  and  z^,  coincide,  and  when  therefore  the  variable 
describes  a  closed  line.  The  above  condition  is  then  changed 
into  this :  the  closed  line  is  not  to  include  any  of  the  critical 
points  mentioned.  Hence,  if  we  let  the  variable  z  starting 
from  Zo  describe  a  closed  line  and  return  again  to  Zq,  the  func- 
tion acquires  here  the  same  value  that  it  had  at  the  beginning, 
if  the  closed  line  include  no  point  at  which  either  discontinuity 
occurs  or  function-values  become  equal. 

Such  closed  lines  described  by  the  variable  z  are  highly 
important  in  the  investigation  of  the  influence  which  the  path 
followed  by  the  variable  z,  on  its  way  to  any  point,  exerts  on 


MULTIFORM  FUNCTIONS.  45 

the  value  which  the  function  w  acquires  at  that  point.  If  a 
closed  line  include  none  of  the  points  already  so  often  men- 
tioned, the  function,  as  has  been  shown,  does  not  change  its 
value;  hut  jf  it  enclose  sij(;>h  ?■  i^oint,  the  function  mav,o^may 
not,  change  its  value.  Further,  if  two  paths  be  described 
by  tne  variable  between  two  points,  which 
enclose  no  point  of  that  kind,  these  lead  to 
the  same  function-value.  Hence  we  have  to 
consider  only  paths  which  enclose  such  a 
point.  Now  let  a  (Fig.  10)  be  a  point  of  this 
kind,  and  assume  two  paths  bdc  and  bee, 
which  enclose  a  but  no  other  similar  point. 
Let  w  start  from  b  with  the  value  ivq  and 
acquire  at  c  the  value  W  along  the  path  bdc. 
Then  if  we  let  the  variable  z,  before  it  enters 

'  Fig.  10. 

on  the  other  path  bee,  describe  a  closed  line 
bghb  round  the  point  a,  the  path  bghbec  can  be  deformed  into 
bdc  without  passing  over  the   point  a;   therefore  w  acquires 
at  c  like\vise  the  value  W  along  this  path,  if  it  start  from  b 
with  the  value  Wq.     We  have  therefore  the  following : 

along    bdc,    w  changes  from  xoq  to  W, 

«    bghbec,  w       «  «     «?„ "  W. 

If  we  first  assume  that  w  changes  its  value  by  the  description 
of  the  closed  line  bghb  and  goes  into  iCi,  we  have : 

along  bghb,  w  changes  from  Wq  to  w^ 

and  hence  "     bee,    w      "  "      w^  "  W. 

Accordingly  w  acquires  at  c  the  value  W  along  bee,  when  it 
starts  from  b  with  the  value  w^ ;  therefore,  if  it  start  from  b 
with  the  value  rco,  it  cannot  acquire  the  value  W,  but  must 
be  led  to  another  value.  If,  on  the  contrary,  w  do  not  change 
its  value  on  the  closed  line  bghb,  we  have : 

along  bghb,  w  changes  from  icq  to  Wq, 

and  hence  "       bee,   lo        "  "      w^ "  W; 


? 


46  THEORY  OF  FUNCTIONS. 

therefore  in  that  case  w,  starting  from  b  with  the  value  zVq, 
acquires  the  value  W  also  along  the  path  bee. 

From  this  follows :  If  two  paths  enclose  one  of  the  points 
a  in  question,  they  lead  to  dilfeient  or  to  the  same  function- 
values,  according  as  the  function  to  does  or  does  not  change 
its  value  in  describing  a  closed  line  round  the  point  a. 

We  are  now  m  a  position  to  define  branch-points  more  pre- 
cisely. A  point  a,  at  which  either  a  discontinuity  occurs  or 
'  several  function-values  become  equal,  is  to  be  called  a  branch- 
point when,  and  only  when,  the  function  changes  its  value  in 
describing  a  closed  line  round  this  (and  no  other  similar)  point. 
Nevertheless,  in  this  connection,  it  is  to  be  noted  that  it  is 
not  necessary  for  all  the  function-values  to  change.  In  order 
that  the  point  in  question  may  be  a  branch-point,  it  is  only 
necessary  for  this  change  to  occur  in  the  case  of  some  one  of  the 
function-values  under  consideration.  For  the  case  can  occur 
that,  in  the  circuit  round  a  branch-point,  only  a  part  of  the 
function-values  change,  while  the  others  remain  unchanged. 
The  example  considered  on  p.  38  ff.  furnishes  such  a  case.  Let 
the  variable  z,  in  Fig.  B,  describe  the  closed  line  dpfmd,  which 

2 
encloses  the  branch-point  e  = ,  then  it  is  evident  from 

V27 

Fig.  A  that  Wa  goes  over  into  W3,  and  Wo  into  w^;  while  w^, 

however,  does  not  change  its  value  but  describes  likewise  a 

closed  line.     Thus  the  proposition  enunciated  at  the  beginning 

of  this  paragraph,  that  two  different  paths   connecting  the 

same  point  assign  different  values  to  a  function,  which  starts 

from  the  initial  point  with  the  same  value,  only  when  they 

enclose  a  branch-point,  is  proved;    and  for  closed  lines,  we 

can   enunciate   the    proposition:    A  multiform   function  can 

pass  from  a  value  corresponding  to  a  point  Zq  to  another  value 

corresponding  to  the  same  point  in  a  continuous  way,  when 

the  variable  z  starting  from  Zq  describes  a  closed  line  which 

encloses  a  branch-point. 

~^^£Uosed  lines,  which  enclose  two  or  more  branch-points,  can 

^.^ewis^  be  reduced  to  such  closed  lines  as  contain  only  one 

branch-point.      For,    if   we   draw   from   a  point   Zq   a   closed 


MULTIFORM  FUNCTIONS.  47 

line  round  each  branch-point  and  let  the  variable  describe  the 
same  in  succession,  then  this  path  can  be  deformed,  without 
passing  over  one  of  the  branch- 
points, into  a  closed  line  which, 
starting  from  Zq,  encloses  all  the 
branch-points.  (Fig.  11,  where 
a  and  b  denote  two  branch- 
points.) We  draw  such  closed 
lines  round  the  individual  branch- 
points most  simply,  by  describ- 
ing round  each  one  a  small  circle, 

and  connecting  each  of  these  circles  with  Zq  by  a  line,  which 
must  then  be  described  twice,  going  and  coming. 

10.  We  will  next  illustrate  the  preceding  considerations 
by  some  examples,  and  at  the  same  time  show  by  them  how 
the  function-values  pass  into  one  another  on  describing  closed 
lines  round  a  branch-point. 

Ex.  1.  w  =  a/z. 

In  this  2  =  0  is  a  branch-point.  If  we  let  the  variable  start 
from  the  point  z  =  1  and  describe  the  circumference  of  a 
circle  round  the  origin,  this  is  a  closed  line  which  encloses 
the  branch-point.  If  the  function  w  =  Vz  start  from  the  point 
z  =  1  with  the  value  w  =  -\-l,  and  if  we  put 

z  =  r  (cos  <{>  +  i  sin  ^), 

then  at  the  point  z  =  l,  r  =  l  and  ^  =  0.  If  z  next  describe 
the  circumference  of  the  circle  in  the  direction  of  increasing 
angles,  r  remains  constant  and  equal  to  1,  and  <(>  increases 
from  0  to  2  TT.     If  therefore  the  variable  return  to  the  point 

z  =  1,  then 

2  =  cos  2ir  +  i  sin  2  tt, 
and  therefore 

w  =  -^  =  cos  TT  -j-  i  sin  tt  =  —  1 ; 

thus  the  function  does  not  now  have  at  the  point  z  =  1  the 
original  value  -j- 1,  but  acquires  the  other  value  —  1.    The  very 


48  THEORY  OF  FUNCTIONS. 

Same  takes  place,  when  the  variable  describes  any  other  closed 
line  once  round  the  origin  starting  from  z  =  l;  for  this  path 
can  be  deformed  into  the  circle  by  gradual  changes  without 
thereby  passing  over  the  origin.  In  general,  if  w  start  with 
the  value  Wq  from  any  point  Zq,  for  which 

Zn  =  n  (cos  <^o  +  i  sin  <^o), 

and  therefore  Wq  =  /*o-  (cos  ^  ^o  +  *  sin  ^  ^o)> 

and  if  z  describe  a  closed  line  once  round  the  origin  in  the 
direction  of  increasing  angles,  then,  on  returning  to  Zq, 

z  =  ro  [cos  («^o  +  2  tt)  +  I  sin  (<^o  +  2  tt)], 

and  therefore  w  =  r^  [cos  {^  ^o  +  tt)  +  i  sin  (^  ^o  +  Tr)] 

=  —  Wfl. 

If  the  variable  describe  the  closed  line  twice,  or  if  it  describe 
another  closed  line  which  winds  round  the  origin  twice,  then 
the  argument  of  z  increases  by  Att,  there- 
fore that  of  w  by  2  tt,  and  consequently  the 
function  then  assumes  again  its  original 
value. 

Now  let  the  variable  go  from  the  point 
2  =  1  to  an  arbitrary  point  Z,  first  along  a 
line  1  eZ  (Fig.  12),  which  does  not  enclose 
the  origin,  and  along  which  the  angles  ^ 
'**■     ■  increase.      Along  this  path  r  and  <^  may 

acquire  at  Z  the  values  B  and  0,  and  w  the  value  W,  so  that 

But  if  the  variable  move  upon  the  other  side  of  the  origin  from 
1  to  Z  along  a  line  1  dZ  not  enclosing  the  origin,  the  angle  <^ 
decreases  and  acquires  at  Z  the  value  ^  —  2  tt.  Hence  at  Z 
in  this  case 

z  =  B  [cos  (2  TT  -  ^)  -  ^•  sin  (2  tt  -  $)], 

and  w  =  B^lcos(7r —  ^6)— ism.(Tr  —  ^&)'] 

=  -W. 


MULTIFORM   FUNCTIONS. 


49 


Finally,  let  z  first  describe  a  closed  line  1 6c  1  round  the  origin 
starting  from  1,  and  next  the  line  1  dZ,  then  <^  first  increases 
from  0  to  2  77  and  next  decreases  by  the  angle  2  tt  —  ^,  so  that 
<fi  acquires  at  Z  the  value  27r  +  ^  —  27r  =  ^;  in  this  case  w, 
after  the  description  of  the  line  1  be  1,  starts  from  1  with 
the  value  —  1  and  acquires  at  Z  the  value  +  W  along  1  dZ. 


Ex.  2.     In  the  function 


(z-l)Vi 


2!  =  0  is  a  branch-point,  and  this  function  behaves  with  respect 
to  this  point  like  the  preceding.  Let  us  consider  therefore 
the  point  z  =  l,  for  which  likewise  w  =  0.  Let  the  variable  z 
describe  round  it  a  circle  with  radius  r,  starting  from  the  point 
a  =  1  +  r  on  the  principal  axis  (Fig.  13).     If  we  put 

z  —  l  =  r  (cos  <f>-\-i  sin  <f>), 

then  w  =  r(cos  (f>  +  ^sin^)Vl  -|-rcos<^  -\-ir8ia<f>. 

Since    r    remains    constant,  and  <^   increases  from  0  to  27r, 
the  factor  r  (cos  <l>  +  i  sin  <^) 
does   not  change   its  value. 
In  order  to  study  the  behav- 
ior of  the  second  factor,  let 

1  +  rcos^  =  pcosij/, 

rsin^  =  psiinf/; 

then  p  denotes  the  straight 
line  oz,  and  if/  the  inclination 
of  the  same  to  the  principal 
axis,  and 

w  =  r  (cos  <^  +  i  sin  ^)  p^(cos  |-  i/f  +  *  sin  \  \f). 

Now,  if  the  circle  do  not  enclose  the  origin,  i/r  passes  through 
a  series  of  values  commencing  with  0  and  ending  with  the  value 
0  again ;  hence  w  does  not  change  its  value.  But  if  the  circle 
be  so  large  that  the  origin,  which  is  a  branch-point,  also  lies 
within  it,  i/^  increases  from  0  to  2  tt,  and  therefore  in  that  case 


Fig.  13. 


60  THEORY  OF  FUNCTIONS. 

the  original  value  w  =  rp^  passes  into  —  rp-.  The  statement 
is  therefore  confirmed,  that  only  the  point  z  =  0  is  a  branch- 
point, and  not  the  point  z  =  l. 

We  can  consider  the  given  function  (2  — l)Vz  as  derived 
from 

w'  =  V(2  —  l){z  —  b)z 

by  making  6  =  1.  A  line  enclosing  the  point  z  =  l  can  then  be 
regarded  as  a  line  which  at  first  enclosed  the  two  points  z  =  l  and 
z=h,  and  in  connection  with  which  these  two  points  were  sub- 
sequently made  to  coincide.  Now  z  =  l  and  z  =  h,  as  well  as 
2  =  0,  are  branch-points  of  the  function  w\  A  closed  line 
which,  starting  from  a  point  Zq,  makes  a  circuit  round  both 
points  1  and  h  can  be  replaced  by  closed  lines,  each  of  which 
encloses  only  one  of  these  points.  If  now  w '  start  from  Zo  with  the 
value  Wo',  on  encircling  the  point  b  it  passes  into  —Wq,  and  then 
on  encircling  the  point  1,  —  Wq  passes  into  Wq'  again.  The 
function  returns  therefore  to  Zq  with  its  original  value.  This 
continues  to  hold  when  b  approaches  the  point  1,  and  when 
these  branch-points  coincide  the  common  point  obviously  ceases 
to  be  a  branch-point.  It  is  evident  that  this  may  be  general- 
ized as  follows :  When  once  in  connection  with  two  branch- 
points only  two  function-values,  and  these  two  the  same,  pass 
mutually  one  into  the  other,  these  branch-points  neutralize 
each  other  on  coinciding,  and  there  arises  a  point  which  is  no 
longer  a  branch-point. 

Ex.  3.     Let  w 


fz-b' 

in  which  a  and  b  denote  two  complex  constants.  In  this  exam- 
ple we  have  two  branch-points,  z  =  a  and  z  =  b.  If  we  first 
let  z  describe  a  closed  line  round  the  point  a  starting  from  an 
arbitrary  point  Zq,  but  not  enclosing  the  point  b,  and  if  we 
accordingly  put 

z  —  a==r  (cos <l>  +  i sin <^), 

while  Zq  —  a  =  r„  (cos  <^o  +  i  sin  <f>o), 

then  the  initial  value  of  w,  which  may  here  be  denoted  by  Wi,  is 


MULTIFORM   FUNCTIONS.  51  ^ 

[a  —  &  +  ro  (cos  (^0  +  ^ sin  <^o)]^  V  ,^"      ' 

After  the  closed  line  is  traversed  once  in  the  direction  dRi  i</ 
increasing  angles,  <^o  lias  increased  by  2  tt,  and  hence  the  resiJjt^  ^ 
ing  value  of  to,  which  will  be  denoted  by  Wo,  is  ^    ^ 

_  n "  [cos  Q-  <^o  +  I  ,r)  +  i  sin  (|  <^o  +  |  ^r)] 

2  2  ■ 

[a  —  6  +  7-0  (cos  ^0  +  i  sin  ^o)]  ^ 

Therein  the  denominator,  and  therefore  the  quantity  -^z—  6, 
cannot  have  changed  its  value,  because  for  it  z  =  a  is  not  a 
branch-point,  but  only  z  =  6 ;  therefore  2  has  described  a  closed 
line  which  does  not  include  the  branch-point  of  this  expression. 
Let 

cos  f  TT  +  t  sin  I  TT  =  — — "^  =  a, 

so  that  a  is  a  root  of  the  equation  a^  =  1 ;  then,  since 

cos  (^  <^o  + 1  it) -j- i  sin  (^  00  + 1  tt) 

=  (cos  ^  <^o  +  «'  sin  \  <^o)  (cos  |  tt  -f  t  sin  |  ir), 
we  can  also  write  w^  =  aiCi. 

Now  let  the  variable  again  describe  a  closed  line  round  the 
point  a ;  then  w  leaves  Zq  with  the  value  Wg  =  awi,  and  there- 
fore acquires,  after  the  completion  of  the  circuit,  the  value 

After  a  third  circuit  w  finally  acquires  the  value  a^Wy,  i.e., 
the  original  value  tOj  again,  since  «^  =  1.  If  we  had  originally 
started  from  Zq  with  the  value  Wj  instead  of  tc\,  we  should  have 
obtained  the  values  Wg  and  Wi  after  one  and  two  circuits  re- 
spectively ;  but  if  Ws  had  been  the  original  value,  this  would 
have  changed  into  tvi  and  it'g  successively. 

Similar  results  are  obtained  when  z  is  made  to  describe  a 
closed  line  including  only  the  point  b.     We  then  put 

z  —  b  =  r  (cos  cf>  -\-i  sin  <f>), 


62  THEORY  OF  FUNCTIONS. 

and  let  w  start  from  Zq  with  the  value  Wi,  this  value  denoting 
the  following  expression : 

^  _[_b  —  a  +  ro  (cos  <^o  +  i  sin  <^o)p 
ro^(cos  I  </>o  +  i  sin  ^  <^o) 

After  one  circuit  by  z  in  the  direction  of  increasing  angles,  the 
value  of  w  becomes 

Ib  —  a  +  rp  (cos  ^p  +  i  sin  <^o)p 
ro^[cos  (i  <^o  +  I  t)  +  i  sin  (|  «/.o  + 1  tt)] 

in  which  now  the  numerator  cannot  have  changed  its  value, 
because  its  branch-point  a  has  not  been  enclosed  in  the  circuit. 
We  therefore  now  obtain  for  w  tlje  value 

—  =  a?Wi,  i.e.,  the  value  Wg. 
a 

After  a  second  circuit  we  obtain 

-i=  awi,  I.e.,  W2', 
a' 

finally,  after  a  third  circuit,  the  original  value  is  restored,  since 

— 3=W?1. 

ay 
It  is  thus  evident  that  the  function-values  for  repeated  cir- 
cuits round  a  branch-point  interchange  in  cyclical  order.    When 
z  moves  round  the  point  a  in  the  direction  of  increasing  angles, 
the  values 

change  after  the  first  circuit  respectively  into 

W2,  W3,  Wi, 

and  after  the  second  circuit  into 

Wg,  Wu  to,; 
after  a  third  circuit  therefore  the  original  values 

Wi,  Wj,  tOa 


MULTIFORM   FUNCTIONS.  53 

are  restored.  In  like  manner,  for  circuits  round  the  point  b 
in  the  direction  of  increasing  angles,  the  values 

pass  into  Wg,  Wi,  Wg, 

and  into  W2,  w^,  Wi, 

and  acquire,  after  the  third  circuit,  the  original  values 

Wj,  W2,  W3. 

Let  us  next  inquire  what  takes  place  when  z  describes  a 
closed  line  including  both  points,  a  and  6.  Such  a  line  can 
always  be  deformed,  without 
passing  over  one  of  these  points, 
into  another  which  consists  of 
successive  circuits  round  them 
(Fig.  11).  Let  then  z  first  de- 
scribe a  circuit  round  the  point 
6  starting  from  Zq,  return  to  z^, 
and  then  describe  a  circuit 
round  the  point  a.  By  this 
path  w  acquires,  on  the  second  return  to  Zq,  the  same  value  as 
when  z  describes  a  closed  line  round  both  branch-points  (§  9). 

If  10  start  from  Zq  with  the  value  Wi,  it  acquires  the  value 

—  =  ^l',  after  the  circuit  round  b,  and  then  after  the  circuit 

round  a  the  value  aw^  =  Wy ;  the  function  reverts  therefore  to 
its  original  value.  If  we  consider  in  this  connection,  instead 
of  the  given  f iijiction,  the  following : 


w'  =  v'(2  —  a)(2  —  b), 

in  which,  as  is  easily  seen,  the  factor  a  is  multiplied  into  the 
original  function-value  after  each  circuit  round  the  point  6; 
then  w\  changes  into  atv'i  =  tc'2  on  making  the  circuit  round  b, 
and  on  making  the  circuit  round  a,  10^2  changes  into  aio'o  =  w'^. 
A  circuit  round  both  points  therefore  changes  w'l  into  «;  3;  hence 
a  second  circuit  will  change  w  3  into  tc's,  and  a  third  w  2  into  w\. 


54  THEORY  OF  FUNCTIONS. 


Ex.  4.   The  function 


w=\ -+Vz--c, 

^Z  —  0 

which  is  the  root  of  the  equation  of  the  sixth  degree, 

{z  -  hfvf  -  3(z  -  h)\z  -  c)w'  -  2(2  -a)(z-  h)w^ 

+ 3(z  -  h)\z  -  c) W  _Q(z-a){z-h){z-c)w 

+  (2  _  of  -{z-  h)\z  -  cy  =  0, 

has  the  branch-points  a,  b,  c.     If,  for  sake  of  brevity,  we  sub- 
stitute 


■\Jz  —  a  —  t,  Vz  —  b  =  u,  V^  —  c  =  V, 

and  give  to  a  the  same  meaning  as  in  the  preceding  example, 
we  can  write  the  six  function-values  as  follows : 

t  ,  t 

u  u 

t  ,  t 

W2=  a  —  -{-v,       Ws=  a V, 

u  u 

Ws  =  (^-  +  V,  Wg  =  «^ V. 

u  u 

Let  us  first  consider  circuits  of  the  variable  round  the  point 
a;  for  these  t  passes  into  at,  aH,  <,•••,  while  u  and  v  remain 
imchanged;  therefore 

iVi  Wg  Wg        w^  w^  Wq 

change  after  the  first      circuit  into      Wg  Wg  w^        w^  Wg  w^ 

"  "       "     second       "        "         WgtUiWg        WsWiW^ 

"  "       "     third  "         "  Wi  w?2  f^a         «'4  ^t's  Wg. 

Eound  this  branch-point,  therefore,  only  the  values  Wj,  W2,  w^ 
permute  by  themselves,  and  W4,  w^,  Wg  by  themselves. 

For  circuits  round  the  point  6,  t  and  v  remain  unchanged, 
and  u  changes  into  au,  o^u,  u,---.     Therefore 


Wl  W2  Wg 

Wi  W5  Wq 

change  after  the  first      circuit  into 

Wg  Wi  V}2 

We  W4  Ws 

"          "      "    second      "        " 

W2  W3  Wi 

W5  We  Wi 

«          «       "     third         "        « 

Wi  W2  Wg 

Wi  Ws  We. 

MULTIFORM   FUNCTIONS. 


55 


Here  the  same  function-values  permute  as  for  tlie  point  a, 
but  in  reverse  sequence. 

Finally,  on  making  circuits  round  the  point  c,  t  and  u  are 
unchanged,  and  v  changes  into  —v,  +  v,  •••.     Hence  here 


change  after  the  first      circuit  into 
u  u      u     second        "       " 


Wi  W2 10^        W4  W5  We 
Wi  W5  We         Wi  W2  Wg 

Wi  W2  IC'g  Wi  Wi  Wg 


In  this  example,  therefore,  we  have  first  two  branch-points, 
round  which  the  three  values  Wj,  102,  w^  permute  in  cyclical 
order,  but  never  with  one  of  the  three  remaining;  likewise, 
Wi,  W5,  iCe  permute  in  cyclical  order  here,  but  never  pass  into 
one  of  the  first  three  values.  We  then  have  one  more  branch- 
point c,  at  which  the  three  pairs  WiWi,  W2Ws,  w^w^,  each  by 
itself,  interchange  their  values,  without  a  value  from  one  pair 
ever  entering  another. 

If  we  let  z  describe  a  closed  line  including  two  branch-points, 
we  can  again  replace  such  a  one  by  two  successive  circuits, 
each  round  one  point.  If  the  points  a  and  6  be  enclosed,  we 
have  the  same  condition  as  in  the  preceding  example.  We 
will  therefore  follow  only  circuits  round  a  and  c,  and  tabulate 
the  results  below. 


CiRCCITS. 

Round  a. 

Bound  e. 

EoiTND  Both. 

1 

wi  changes  into  xp2 

W2  into  W5 

Wi into  wg 

2 

W5       "          "    We 

We    "    Ws 

W5      "      Ws 

3 

W3          "              "      Wi 

Wl      "      lOi 

Ws      "      W4 

4 

Wi          "              "      Ws 

Ws      "      W2 

W4      "      t02 

5 

W2          "              "      Ws 

W3    "    We 

W2    "    We 

6 

We       "          "     Wi 

W4      "      Wi 

We    "    tci 

Therein  w  acquires  its  original  value  only  after  six  consecu- 
tive circuits  round  the  points  a  and  c. 

11.   The  preceding  considerations  show  that,  given  a  multi- 
form function,  we  can  pass  continuously  from  one  of  the  values 


66  THEORY  OF  FUNCTIONS. 

■which  the  function  can  assume  for  the  same  value  of  the  vari- 
able to  another,  by  assigning  complex  values  to  the  variable 
and  letting  it  pass  through  a  series  of  continuously  successive 
values,  which  ends  with  the  same  value  with  which  it  began 
(geometrically  expressed,  by  letting  the  variable  describe  a 
closed  line).  It  has  been  further  shown  that  a  definite  and 
continuous  series  of  values  of  the  variable  (a  definite  path), 
also,  always  leads  to  a  definite  function-value,  except  in  the 
single  case  when  the  path  of  the  variable  leads  through  a 
branch-point,  a  case,  however,  which  can  always  be  obviated 
by  letting  the  variable  make  an  indefinitely  small  deviation  in 
the  vicinity  of  the  branch-point.^  This  naturally  suggests  the 
desirability  of  avoiding  the  multiplicity  of  values  of  a  multi- 
form function,  in  order  to  be  able  to  treat  such  a  function  as  if 
it  were  uniform.  According  to  the  preceding  explanations,  it  is 
necessary  for  this  purpose  only  to  do  away  with  the  multiplicity 
of  paths  which  the  variable  can  describe  between  two  given 
points.  Now  Cauchy  has  already  remarked  that  thi  s  could  be  • 
effected,  at  least  to  a  limited  extent,  by  demarcating  certain 
portions  of  the  plane  in  which  the  variable  z  is  supposed  to  be 
moving  and  not  permitting  the  latter  to  cross  the  boundary  of 
such  a  region.  For,  since  a  function,  starting  from  a  point  Zq 
of  the  variable,  can  assume  different  values  at  another  point  Zi, 
only  when  two  paths  described  by  the  variable  enclose  a 
branch-point  (§  9),  it  is  always  easy  to  mark  off  a  portion  of 
the  z-plane  within  which  two  such  paths  from  Zq  to  Zi  are  not 
possible,  or,  by  drawing  certain  lines  which  start  from  branch- 
points, and  which  are  not  to  be  crossed,  to  make  such  paths 
impossible.  Within  such  a  region  the  function  remains  uni- 
form, since  it  acquires  at  each  point  z^  only  a  single  value  along 
all  paths.  The  function  is  then  called  monodromic  (after 
Cauchy)  or  uniform,  one- valued  (after  Riemann).  Although 
this  method  is  of  great  advantage,  for  instance,  in  the  evaluar 

1  If  we  regard  the  position  of  the  path  of  the  variable,  in  case  it  lead 
through  a  branch-point,  as  the  limiting  position  of  a  path  not  meeting  the 
brancli-point,  then  to  this  assumption  corresponds  again  a  definite  func- 
tion-value. 


MULTIFOBM  FUNCTIONS.  57 

tion  of  definite  integrals,  nevertheless  by  means  of  it  only  a 
definite  region  of  values,  or,  as  Riemann  calls  it,  a  definite 
branch  of  the  multiform  function,  is  separated  from  the  rest 
and  considered  by  itself.  In  order  to  be  able  to  treat  an  alge- 
braic function  in  its  entirety  and  yet  as  if  it  were  a  uniform 
function,  Riemann  has  devised  another  method,  which  will  be 
set  forth  in  the  following. 

Riemann  assumes  that,  when  a  function  is  n-valued,  when 
therefore  to  every  value  of  the  variable  n  values  of  the  function 
correspond,  the  plane  of  z  consists  of  n  sheets  or  leaves  (or  that 
n  such  sheets  are  extended  over  the  z-plane),  which  together 
form  the  region  for  the  variable.  To  each  point  in  each  sheet 
corresponds  only  a  single  value  of  the  function,  and  to  the  n 
points  lying  one  immediately  below  another  in  all  the  n  sheets 
correspond  the  n  different  values  of  the  function  which  belong 
to  the  same  value  of  z.  ISTow  at  the  branch-points,  where  sev- 
eral function-values,  elsewhere  different,  become  equal  to  one 
another,  several  of  those  sheets  are  connected,  so  that  the  par- 
ticular branch-point  is  supposed  to  lie  at  the  same  time  in  all 
these  connected  sheets.  The  number  of  these  sheets  thus  con- 
nected at  a  branch-point  can  be  different  at  each  branch-point, 
and  is  equal  to  the  number  of  function  values  which  change 
one  into  another  in  cyclical  order  for  a  circuit  of  the  variable 
round  the  branch-point.  In  the  last  example  of  the  preceding 
paragraph,  wherein  the  function  is  six-valued,  we  shall  assume 
the  2-plane  as  consisting  of  six  sheets.  Round  each  of  the 
branch-points  a  and  b  the  values  ic^,  w.2,  w^,  on  the  one  hand, 
and  ?(-'4,  W5,  Wq,  on  the  other,  change  one  into  another ;  hence  we 
assume  that  at  each  of  these  points  the  sheets  1,  2,  3,  on  the 
one  hand,  and  the  sheets  4,  o,  6,  on  the  other,  are  connected. 
Round  the  point  c,  however,  firstly,  iVi  and  w^ ;  secondly,  t^g  and 
Ws ;  thirdly,  tv^  ^^^  *^6>  change  respectively  one  into  the  other ; 
hence  at  the  point  c,  first  the  sheets  1  and  4,  then  the  sheets 
2  and  5,  and  finally  the  sheets  3  and  6  are  connected.  Now 
for  the  purpose  of  exhibiting  the  continuous  change  of  one 
value  of  the  function  into  another,  so-called  branch^cuts^  ave 
1  Sometimes  called  cross-lines.     (Tr.) 


68 


THEORY  OF  FUNCTIONS. 


introduced.  These  are  quite  arbitrary  lines  (except  that  one 
cannot  intersect  itself),  which  either  pass  from  a  branch-point 
to  infinity,  or  join  with  each  other  two  branch-points.  We  do 
not  suppose  the  sheets  to  be  connected  along  these  branch-cuts 
as  they  naturally  lie  one  above  another,  but  as  the  function- 
values  interchange  round  the  respective  branch-points.  If,  for 
instance,  in  the  last  example  of  the  preceding  paragraph,  we 
draw  a  branch-cut  from  a  to  6  (Fig.  14),  we  then,  in  making  a 
circuit  round  the  point  a  in  the  direction  of  increasing  angles, 
connect  the  sheet  1  with  the  sheet  2  along  the  branch-cut,  then 
2  with  3,  and  finally  3  with  1  again.  Let  us  call  the  right  side 
of  the  branch-cut  ab,  that  which  an  observer  has  on  his  right, 
when  he  stands  at  a  and  looks  toward  b.  Then  if  z  go  from  a 
point  Zq  in  sheet  1  (w  with  the  value  Wi)  and  make  a  circuit 


VS5: 


"3-A. 


Fig.  14. 


round  the  point  a  in  the  direction  of  increasing  angles,  on 
crossing  the  branch-cut  from  right  to  left,  it  passes  from  the 
first  sheet  into  the  second  and  is  still  in  the  latter  when  it 
returns  to  Zq,  or  rather  to  the  point  g  lying  immediately  below 
Zq  in  the  second  sheet,  so  that  w  acquires  the  value  Wg  If 
the  description  be  still  continued,  z  passes,  after  crossing  the 
branch-cut  a  second  time  from  right  to  left,  into  the  third  sheet 
and  is  still  in  the  same  when  it  arrives  at  the  point  h  situ- 
ated in  this  sheet  below  Zq  ;  w  has  now  acquired  the  value  w^. 
Finally,  when  z  crosses  the  branch-cut  a  third  time,  we  assume 
that  the  right  side  of  the  third  sheet  is  connected  along  the 
branch-cut  with  the  left  side  of  the  first  sheet  through  the 
second  sheet,  so  that  z  crosses  from  the  third  sheet  to  the  first 
sheet  and  then  returns  again  to  Zq. 


MULTIFORM  FUNCTIONS.  59 

Not  until  now  is  the  line  actually  closed,  and  has  w  acquired 
again  its  original  value.  In  Fig.  14  the  lines  are  denoted  by 
the  numbers  of  the  sheets  in  which  they  run,  and,  in  addition, 
those  in  the  second  and  third  sheets  are  thickly  dotted  and 
thinly  dotted  respectively.  The  points  Zq,  g,  h,  which  ought 
really  to  be  one  directly  below  another,  are,  for  the  sake  of 
clearness,  drawn  side  by  side. 

A  similar  course  must  be  imagined  in  the  case  of  all  branch- 
points, and  since  from  each  such  point  a  branch-cut  starts,  the 
variable  cannot  make  a  circuit  round  the  branch-point  without 
crossing  the  branch-cut,  and  thereby  passing  in  succession 
into  all  those  sheets  which  are  connected  at  the  branch-point. 
How  the  branch-cut  is  to  be  drawn  in  each  case  depends 
upon  the  function  to  be  investigated,  and  can  generally  be 
chosen  in  different  ways.  In  our  example  a  and  b  may  be 
connected  by  such  a  cut,  because  with  circuits  round  the  point 
b  in  the  direction  of  increasing  angles  the  function  Wy  changes 
into  zvs  and  this  into  Wg  (Fig.  14),  and  hence  at  6  the  same 
sheets  are  connected  as  at  a,  and  in  the  same  way ;  namely, 
the  right  side  of  1  with  the  left  of  2,  the  right  side  of  2  with 
the  left  of  3,  and  the  right  side  of  3  with  the  left  of  1.^ 

Let  us  continue  with  this  example  and  investigate  the  cir- 

1  If  we  wish  to  exhibit  this  method  of  representation  by  a  model,  a  diffi- 
culty arises,  first,  because  the  sheets  of  the  surface  interpenetrate,  and 
in  the  second  place,  because  frequently  at  branch-points  several  sheets, 
which  do  not  lie  one  immediately  below  another,  must  be  supposed  to 
be  connected.  But  for  the  purpose  of  illustration,  it  is  for  the  most  part 
necessary  only  to  be  able  to  follow  certain  lines  in  their  course  through 
the  different  sheets  of  the  surface.  This  can  be  easily  effected  in  the 
following  way  :  First  cut  in  the  sheets  of  paper  placed  one  above  another, 
which  are  to  represent  the  surface,  the  branch-cuts,  and  then  only  at 
those  places  where  a  line  is  to  pass  over  a  branch-cut  from  one  sheet 
into  another  join  the  respective  sheets  by  pasting  on  strips  of  paper. 
Then  we  can  always  so  contrive  that,  when  the  line  is  to  return  to  the 
first  sheet,  from  which  it  started,  we  have  the  necessary  space  left  for 
the  fastening  of  the  strip  of  paper  by  means  of  which  the  return  passage 
is  effected.  By  these  attached  paper  strips  union  of  the  separate  sheets 
into  one  connected  surface  is  accomplished ;  and  it  is  then  no  longer 
necessary  to  connect  the  sheets  with  one  another  at  the  branch-points. 


60 


THEORY  OF  FUNCTIONS. 


cuits  discussed  in  the  preceding  paragraph  round  a  and  h,  and 
round  a  and  c.  For  a  circuit  round  a  and  h  the  branch-cut 
is  not  crossed  at  all,  so  that  z  remains  in  the  first  sheet;  in 
fact  w  resumes  its  original  value  at  Zq  after  such  a  circuit. 
(Cf  Ex.  3,  §  10).  To  examine  the  circuit  round  the  points  a 
and  c,  let  us  draw  from  c  to  infinity  a  branch-cut,  and  let  here 
the  sheets  of  every  pair  1,  4 ;  2,  5 ;  3,  6  pass  respectively  into 
each  other. 

For  the  passages  of  the  function-values  taking  place  here, 
we  have  found  the  following  table  (p.  55) : 


Circuits. 

EoirsD  a. 

EOCND   C. 

Round  Both. 

1 

Wi  changes  into  ic^ 

W2  into  iC6 

lOi  into  ws 

2 

Ws         "            "      Wg 

We    "    W3 

t05      "      Ws 

3 

103          "              "      Wi 

tPi      "      tC4 

Wa    "    W4 

4 

W4          "              "      Wb 

Ws      "      W2 

W4      "      W2 

5 

%0i          "              "      Wz 

ws    "    tce 

W2    "    We 

6 

We       "          "    tP4 

W4      '*      Wi 

iCe    "    wi 

Fig.  15. 


These  passages  are  represented  in  Fig.  15,  by  designating 
each  line  by  the  number  of  the  sheet  in  which  it  runs.  The 
points  properly  lying  below  the  initial  point  1  are  represented 


MULTIFORM  FUNCTIONS.  61 

side  by  side,  for  the  sake  of  clearness,  and  the  last  point  1  is 
to  be  supposed  to  coincide  with  the  first. 

This  region  for  the  variable  z,  consisting  in  our  example  of 
six  sheets,  forms  a  single  connected  surface,  since  the  sheets 
are  connected  at  the  branch-points  and  pass  into  one  another 
along  the  branch-cuts.  In  this  surface  w  is  a  perfectly  uni- 
form function  of  the  position  in  the  surface,  since  it  acquires 
the  same  value  at  every  point  of  the  latter,  along  whatever 
path  the  variable  may  arrive  at  the  point.  If  z  describe 
between  two  points  two  paths,  which  enclose  a  branch-point, 
then  one  of  the  two  must  necessarily  cross  a  branch-cut  and 
thereby  pass  into  another  sheet,  so  that  the  terminal  points 
of  the  two  paths  are  no  longer  to  be  considered  coincident, 
but  as  two  different  points  of  the  2-surface,  at  which  different 
function-values  occur.  But  if  z  describe  an  actually  closed 
curve,  i.e.,  if  the  initial  and  the  terminal  points  of  the  curve 
coincide  in  the  same  point  of  the  same  sheet,  then  also  the 
function  acquires  its  initial  value.  Only  when  the  variable 
passes  through  a  branch-point  can  it  pass  at  will  into  any  one 
of  the  sheets  connected  at  that  point,  and  in  that  case  it 
remains  undetermined  which  value  the  function  assumes  (§  8). 

12.  Now  in  order  to  prove  that  we  can  also,  in  general, 
transform  an  7i-valued  function  into  a  one-valued  by  means  of 
an  n-fold  surface  covering  the  ^-plane,  the  single  sheets  of 
which  are  connected  at  the  branch-points  and  along  the  branch- 
cuts  in  the  manner  explained  above,  we  first  assume  the  z-plane 
to  be  still  single,  and  let  the  variable  z,  starting  from  an  arbi- 
trary point  Zq,  describe  a  closed  line,  which  makes  a  circuit 
once  round  a  single  branch-point  and  does  not  pass  through 
any  other  branch-point.  At  Zq  the  function  possesses  n  values ; 
let  us  assume  them  to  be  written  down  in  a  certain  sequence. 
Now  after  the  variable  has  described  the  closed  line  and  re- 
turned again  to  z,),  each  of  the  above  n  function-values  will 
have  either  passed  into  another  or  remained  unchanged.  Since 
the  variable  z  is  again  at  the  point  Zq,  these  new  values  of  the 
function  cannot  differ  from  the  former  in  their  totality;  but 


62  THEORY  OF  FUNCTIONS. 

if  we  suppose  them  to  be  written  down  in  the  sequence  in 
which  they  have  arisen  in  succession  from  the  former,  they 
will  occur  in  an  arrangement  different  from  the  previous  one. 

Any  one  arrangement  of  n  elements,  however,  can  be  derived 
from  another  arrangement  by  a  series  of  cyclical  permutations. 
By  a  cyclical  permutation  of  the  pth  order  we  understand  one 
in  which  from  the  existing  n  elements  we  take  out  p  arbitrarily 
and  in  the  place  of  the  first  of  these  put  a  second,  in  the  place 
of  the  second  a  third,  etc.,  and  finally  in  the  place  of  the  pth. 
the  first.  Such  a  cyclical  permutation  of  the  pth  order  has  the 
property,  that  after  p  repetitions  of  it,  and  not  sooner,  the 
original  arrangement  is  restored ;  for,  since  the  place  of  each 
element  is  taken  by  another,  that  of  the  j9th  element,  however, 
by  the  first,  each  element  can  reappear  in  its  original  place 
only  after  all  the p—1  other  elements  have  occupied  the  same 
place;  then,  however,  each  element  actually  returns  to  its 
original  place.  Now,  to  prove  that  each  arrangement  can  be 
derived  from  another  by  a  series  of  cyclical  permutations,  we 
assume  that  any  one  arrangement  arises  from  another  by  sub- 
stituting one  element,  say  3,  for  another,  say  1.  The  place  of 
3  is  then  taken  either  by  1,  in  which  case  we  have  already  a 
cyclical  permutation  of  the  second  order,  or  by  some  other  ele- 
ment, say  5.  The  place  of  the  latter  is  then  again  taken  either 
by  the  first,  1,  in  which  case  we  have  a  cyclical  permutation  of 
the  third  order,  or  again  by  another  which  is  different  from  those 
already  employed,  1, 3,  5.  Its  place  can  be  taken  either  by  the 
first,  whereby  a  cyclical  permutation  would  be  closed,  or  again 
by  another;  finally,  however,  the  cyclical  permutation  must 
terminate,  because  altogether  there  is  only  a  finite  number  of 
elements,  and  the  first  element  1  must  be  found  in  some  place 
of  the  second  arrangement.  In  this  way  then  a  series  of  ele- 
ments is  disposed  of.  If  we  next  begin  with  some  one  of  the 
elements  not  yet  employed,  we  can  repeat  the  former  procedure 
until  all  the  elements  have  been  exhausted,  and  we  thus  obtain 
a  certain  number  of  cyclical  permutations  which,  employed 
either  successively  or  also  simultaneously,  produce  the  second 
arrangement  from  the  first.     If  an  element  have  not  changed 


MULTIFORM  FUNCTIONS.  63 

its  place  in  the  second  arrangement,  such  a  permanence  can  be 
regarded  as  a  cyclical  permutation  of  the  first  order.  Let  us 
illustrate  the  above  by  an  example.     Suppose  the  elements 

123456789    10    11 

have  passed  into  the  arrangement 

3    11     527     10    19684; 

it  is  evident  that  the  row 

13    5    7 

has  changed  into  3    5     7    1; 

these  therefore  form  a  cyclical  permutation  of  the  fourth 
order.     If  we  next  proceed  from  2,  it  is  evident  that  the  row 

2    11     4 

changes  into  11      4    2; 

therefore  we  have  a  second  cyclical  permutation,  of  the  third 
order.     The  next  element  not  yet  employed  is  6.     Then  the 

row  6    10    8    9 

changes  into  10      8    9     6, 

and  we  have  a  third  cyclical  permutation,  of  the  fourth  order. 
Now  all  11  elements  are  exhausted,  and  consequently  the 
second  given  arrangement  is  derived  from  the  first  by  the 
three  cyclical  permutations  obtained  above. 

If  we  now  return  to  our  function-values,  it  follows  that, 
whatever  arrangement  of  them  may  have  arisen  from  the  cir- 
cuit of  the  variable  round  the  branch-point,  it  can  be  produced 
by  a  series  of  cyclical  permutations  of  the  function-values.  If 
the  variable  be  made  to  describe  a  circuit  round  the  same 
branch-point  a  second  time,  each  function-value  undergoes  the 
same  change  that  it  experienced  the  first  time.  For  this 
second  circuit,  therefore,  the  cycles  remain  the  same  as  for  the 
first,  and  so,  too,  for  each  subsequent  circuit.     Thus  the  values 


64  THEORY  OF  FUNCTIONS. 

of  the  function  (unless  they  all  form  a  single  cycle,  which  case 
can  also  occur ;  cf .  Ex.  3,  §  10)  are  divided  at  each  branch-point 
into  a  series  of  cycles,  so  that  in  each  cycle  only  certain 
definite  values  of  the  function  can  permute  among  themselves 
with  the  total  exclusion  of  all  values  contained  in  another 
cycle  (cf.  Ex.  4,  §  10). 

If  a  single  value  of  the  function  do  not  change  for  the  cir- 
cuit of  the  variable  round  the  branch-point,  the  same  can  be 
regarded,  as  has  been  remarked  above,  as  forming  by  itself 
a  cycle  of  the  first  order.  If  the  variable  be  now  made  to 
describe  some  quite  arbitrary  closed  line,  the  latter  can  be 
deformed  into  a  series  of  circuits  round  single  branch-points 
(§  9).  Therefore  the  arrangement  arising  through  this  closed 
line  can  also  be  produced  by  the  cyclical  permutations  occur- 
ring at  the  branch-points. 

If,  now,  the  z-surface  be  supposed  to  consist  of  n  sheets,  the 
preceding  justifies  the  assumption  that,  at  each  branch-point, 
certain  sets  of  sheets  are  supposed  to  be  connected,  which  con- 
tinue into  one  another  along  the  branch-cuts  in  the  way  above 
stated.  Then  the  variable,  by  describing  a  circuit  round  a 
branch-point,  passes  in  succession  into  all  those  sheets  belong- 
ing to  the  same  group,  and  into  none  but  those,  and  finally  it 
returns  into  that  sheet  from  which  it  started. 

To  find  for  an  w-valued  algebraic  function  the  Riemann 
n-sheeted  surface,  let  us  first  determine  its  branch-points  and 
choose  some  definite  value  Zq  of  z,  which,  however,  must  not 
itself  be  a  branch-point.  We  must  then  let  z  describe  a  cir- 
cuit in  the  single  2;-plane  once  round  each  separate  branch- 
point, always  starting  from  Zq  and  returning  again  to  it,  and 
we  must  ascertain  how  the  function-values  occurring  at  Zq 
are  divided  at  each  branch-point  into  the  above-mentioned 
cycles,  and  how,  within  these,  they  permute  with  one  another.^ 

1  On  this  point  cf.  Puiseux,  "  Recherches  sur  les  fonctions  algfibriques," 
Liouville,  Journ.  de  Math.  T.  xv.  (In  the  German  by  H,  Fischer,  Puiseux'' s 
Untersuchungen  uber  die  algehraischen  Funktionen.  Halle,  1861.)  Konigs- 
berger,  Vorlesungen  uber  die  Theorie  der  ell.  Funktionen.  Leipzig,  1874. 
I.  S.  181. 


MULTIFOBM  FUNCTIONS.  65 

This  having  been  ascertained,  if  in  the  w-sheeted  surface  the 
points  of  the  ?j,-sheets  which  represent  the  value  Zq  be  desig- 
nated in  succession  by  z°,  z.°,  •••,  z°,  so  that  the  subscript 
indicates  the  sheet  in  which  the  point  is  situated,  then  we 
can  first  arbitrarily  distribute  the  values  of  the  function  at 
Zq  among  the  sheets ;  i.e.,  we  can  assume  in  an  arbitrary  but 
definite  way  which  of  these  values  of  the  function  shall  belong 
to  each  of  the  points  z^,  z.°,"-,z°.  We  will  denote  these 
values  in  order  by  w°,  iv^,  •••,  w„°-  Let  us  next  draw  from 
each  branch-point  a  branch-cut  to  infinity,  and  for  each  of  the 
latter  let  us  so  determine  the  connection  of  the  sheets  that  it 
shall  exactly  correspond  to  the  cycles  previously  ascertained. 
If,  therefore,  in  the  single  2^plane,  for  a  single  circuit  round  a 
certain  branch-point,  w°  be  changed  into  t<;^°,  w°  into  w°,  etc., 
then  in  the  ^-sheeted  surface  the  connection  of  the  sheets  is 
so  determined  that  for  a  single  circuit  round  the  same  branch- 
point, the  variable  passes  from  z°  to  z^,  from  z°  to  z^,  etc. 
If  a  single  function-value  w^  suffer  no  change  thereby,  the 
corresponding  sheet  /u.  is  connected  with  no  other  sheet,  so 
that  the  sheet  fi  continues  into  itself  along  the  branch-cut,  it 
being  therefore  unnecessary  to  draw  the  branch-cut  in  this 
sheet. 

If,  as  has  been  assiimed,  a  branch-cut  extend  from  each 
branch-point  to  infinity,  the  connection  of  the  sheets  can  be 
determined  at  each  branch-point  independently  of  the  others. 
This,  however,  does  not  exclude  the  possibility  of  sometimes 
connecting  two  branch-points  by  a  branch-cut,  or  of  making 
several  branch-cuts  extend  from  one  branch-point;  but  this 
may  occur  only  when  the  previously  ascertained  way  in  which 
the  function-values  permute  at  the  respective  branch-points 
permits  such  an  arrangement.  Thus,  in  the  example  of  a  six- 
valued  function  considered  in  the  preceding  paragraph,  a  branch- 
cut  can  be  drawn  from  each  of  the  three  branch-points  a,  b,  c 
to  infinity;  but  the  way  in  which  the  values  of  the  fimction 
permute  round  a  and  b  also  admits  of  a  and  b  being  connected 
by  a  branch-cut. 

These   determinations   having  once  been    established,    the 


66  THEORY  OF  FUNCTIONS. 

function-values  for  each  value  of  z  are  distributed  among  the 
n  sheets  in  a  definite  way.  To  prove  this,  since  in  the  single 
2-plane  the  different  values  which  the  function  can  have  at  one 
and  the  same  point  z  are  produced  only  by  the  different  paths 
to  z,  it  is  only  necessary  to  show  that,  if  in  the  ?i-sheeted  sur- 
face starting  from  a  certain  point,  say  2:1°,  and  with  the  definite 
value  Wi°,  we  reach  the  same  arbitrarily  chosen  point  z^  along 
any  two  different  paths,  we  are  always  led  to  the  same  function- 
value  (by  the  two  paths).  In  this  proof  different  cases  must 
be  distinguished. 

(1)  Let  us  first  assume  that  the  terminal  point  of  a  path 
starting  from  2:1°  is  one  of  the  points  representing  the  value  Zq, 
say  z^°.  Then  the  corresponding  path  in  the  single  z-plane 
forms  a  closed  line.  This  can  be  deformed  into  a  series  of 
(closed)  circuits  round  single  branch-points  without  changing 
the  final  value  of  the  function.  Corresponding  to  this  in  the 
7i-sheeted  surface,  the  variable  also  makes  circuits  round  single 
branch-points  and  after  each  circuit  goes  to  whichever  of  the 
points  z°,  Z2°,  •••,  z„°  it  can  reach.  The  points  at  which  it  arrives 
in  this  manner  may  be  designated  in  order  by  Zi°,  z°  z^,  •••, 
z°,  z^°.  According  to  the  principles  established  above  concern- 
ing the  arrangement  of  the  function-values,  and  since  here  only 
single  circuits  round  any  one  of  the  branch-points  are  taken 
into  consideration,  it  follows  that  w  assumes  in  succession  the 
values  w°,  w°,  tVp°,  •••,  wj",  w^°.  Now  a  similar  deformation 
can  be  made  in  the  case  of  any  other  path  starting  from  Zi°  and 
terminating  in  z^°,  although  then  the  variable  "wdll  generally 
pass  into  other  sheets  than  before.  Let  us  assume  that  the 
circuits  lead  the  variable  in  succession  from  Zi°  to  z/,  Zj°,  •••,  z/, 
and  finally  to  z^°,  then  the  corresponding  series  of  function- 
values  is  w°,  w„°,  Wj,°,  •••,  w^°,  and  since  the  last  circuit,  accord- 
ing to  the  assumption,  leads  from  z/  to  z^°,  w^°  must  finally  also 
change  into  iv^°.     This  may  be  illustrated  by  an  example. 

In  the  six-sheeted  surface  represented  in  §  11  we  can  reach 
for  instance  Zg"  from  z°  by  two  circuits  round  the  point  a,  by 
which  we  come  first  from  Zi°  to  Z2°,  and  then  to  Zg".  But  among 
others  we  can  also  choose  the  following  way :  from  z°  round 


MULTIFORM  FUNCTIONS.  67 

a  to  Z2°,  then  round  c  to  2:5°,  then  round  a  to  Ze°,  and  finally 
round  c  to  2:3°.  For  the  first  path  w  assumes  the  values  Wi°,  lo^, 
w^ ;  for  the  second,  11-°,  w^,  w^,  Wq°,  w^,  but  the  final  value  is 
the  same  for  both  paths.  It  must  be  emphasized  as  a  special 
case  that,  if  the  variable  return  by  any  one  path  to  its  starting- 
point  z°,  the  function  also  resumes  the  original  value  w°. 

(2)  We  next  consider  two  paths  A  and  B  connecting  the 
same  points  and  running  entirely  in  one  and  the  same  sheet ; 
therefore,  if  for  instance,  we  assume  again  21°  as  the  starting- 
point,  entirely  in  the  sheet  1.  Then,  if  the  two  paths  enclose 
no  branch-point,  no  special  discussion  is  required,  since  the  cor- 
responding paths  in  the  single  z-plane  also  enclose  no  branch- 
point, and  therefore  lead  to  the  same  value  of  the  function 
(§  9).  It  is  to  be  noted  that  this  case  can  occur,  if  the  paths 
considered  enclose  branch-points,  only  when  from  neither 
of  the  enclosed  branch-points  a  branch-cut  extends  to  infinity ; 
for,  otherwise,  at  least  one  of  the  paths  would  have  to  cross 
the  branch-cut  and  could  not  run  entirely  in  the  same  sheet. 
The  case  under  consideration  can  therefore  only  occur  when 
several  branch-points  are  enclosed  by  the  two  paths,  and  when 
each  branch-cut  connects  two  branch-points  with  each  other.  If 
then,  in  the  single  2:-plane,  we  let  circuits  round  all  the  enclosed 
branch-points  precede  one  of  the  paths,  say  B,  we  obtain  a  new 
path  C  leading  to  the  same  function-value  as  does  A.  But  if 
these  circuits  be  made  in  the  n-sheeted  surface,  they  again  lead 
back  to  z°,  because  each  branch-cut  connects  two  branch-points, 
and  therefore,  for  the  successive  circuits  round  the  latter,  the 
branch-cut  must  be  crossed  twice  in  opposite  directions;  we 
thus  always  come  back  to  the  sheet  1,  and  therefore  finally 
also  to  2:1°.  Then,  according  to  what  has  been  proved  in  the 
preceding  case,  the  function  also  acquires  again  at  z°  the  value 
Wi.  Since,  however,  the  path  C,  which  consists  of  the  circuits 
and  the  path  B,  leads  to  the  same  value  as  does  A,  and  since 
the  function  starts  on  the  path  B  with  the  value  w°,  therefore 
this  alone  must  also  lead  to  the  same  value  as  does  A. 

(3)  Finally,  let  us  assume  as  the  terminal  point  of  the  paths 
to  be  examined  any  point  z^^  lying  in  any  arbitrary  sheet  A..    Let 


68  THEORY  OF  FUNCTIONS. 

the  initial  point,  as  before,  be  2:1°.  For  sucli  a  path  we  can,  in 
the  first  place,  without  changing  the  final  function-value,  sub- 
stitute another  path,  which  first  leads  to  z^^  and  then,  running 
entirely  in  the  sheet  A,  to  z^^ ;  for,  in  the  corresponding  paths  in 
the  single  z-plane,  the  terminal  portion  of  the  second  can  always 
be  so  chosen  that  this  can  be  deformed  into  the  first  without 
necessarily  crossing  a  branch-point.  If  we  now  make  the 
same  deformation  in  the  two  different  paths,  both  first  lead  to 
Z)^;  here  the  function  according  to  (1)  acquires  along  both 
paths  the  value  W;^°.  The  portions  of  the  two  paths  still  remain- 
ing run  entirely  in  the  sheet  X,  commence  at  the  same  point  z^^ 
and  with  the  same  value  of  the  function  W)^ ;  therefore,  both 
according  to  (2)  also  lead  to  the  same  value  of  the  function  at  z^. 

We  have  thus  proved  that  after  the  above  determinations, 
chosen  quite  arbitrarily,  have  been  made,  the  function  acquires 
at  each  point  of  the  surface  a  definite  value  independent  of  the 
path  and  becomes  a  one-valued  function  of  the  position  in  the 
surface.  Thereby  we  have  removed  the  multiformity  of  alge- 
braic functions,^  and  in  what  follows  we  shall  now  always 
assume  that  the  region  of  the  variable  consists  of  as  many 
sheets  as  are  required  to  change  the  multiform  function  under 
consideration  into  a  uniform  one,  and  we  shall  consider  two 
points  as  identical,  only  when  they  also  belong  to  the  same 
sheet  of  the  surface. 

Accordingly  we  shall  call  a  line  actually  closed,  only  when 
its  initial  and  terminal  points  coincide  at  the  same  point  of  the 
same  sheet.  On  the  other  hand,  should  a  line  end  in  a  point 
situated  in  another  sheet  above  or  below  the  initial  point,  we 
shall  sometimes  call  such  a  line  apparently  closed. 

13.  To  the  above  considerations  let  us  add  a  few  remarks. 
In  crossing  a  branch-cut,  one  sheet  is  continued  into  another, 

1  Though  we  can  also  ascribe  branch-points  to  such  functions  as  log  z, 
tan"iz,  etc.,  we  should  then  be  obliged  to  assume  that  an  infinite  number 
of  sheets  of  the  surface  are  connected  in  a  branch-point.  For  this  reason 
the  functions  mentioned  will  be  considered  later  from  the  point  of  view 
of  functions  defined  by  integrals. 


MULTIFORM  FUNCTIONS.  69 

as  has  been  set  forth  above,  in  such  a  manner  that,  when  the 
variable  moves  in  it,  the  function  changes  continuously.  It 
follows  from  this,  which  is  to  be  well  heeded,  that  the  function 
in  the  same  sheet  must  always  have  different  values  on  the  two 
sides  of  a  branch-cut.  Let  us  assume,  for  example,  that  a  sheet 
K  is  continued  beyond  a  branch-cut  into  another  sheet  X,  and 
let  z^  and  z^  be  two  points  representing  the  same  value  of  z, 
and  lying  in  k  and  A.  respectively  and  infinitely  near  the  branch- 
cut.  Further,  let  z'^  be  the  point  which  lies  in  k  on  the  other 
side  of  the  branch-cut  directly  opposite  z^  and  infinitely  near 
the  same.  Then  the  variable,  in  order  to  pass  from  z^  to  z^, 
must  first  move  round  the  branch-point  to  z'^,  from  which  point 
it  immediately  comes  to  z^  by  crossing  the  branch-cut.  Accord- 
ingly z'^  and  z^  are  in  continuous  succession ;  z^  and  z^,  however, 
are  not.  If  w^,  w'^,  w^  denote  the  function-values  corresponding 
to  z^,  z'^,  Zj^  respectively,  then  u\  is  continuous  with  w'^,  but  not 
with  ti\,  and  if  we  disregard  the  infinitely  small  difference 
between  tc\  and  w'^,  we  can  say  w'^  =  ic^^ ;  but  since  tt\  is  differ- 
ent from  w^,  w^  and  w'^  are  also  different  from  each  other. 
Take,  for  instance,  the  function  iv  =  Vz.  The  surface  then 
consists  of  two  sheets,  which  are  connected  at  the  branch-point 
z  =  0.  Here  w^  =  —  w^  (cf.  §  10,  Ex.  1) ;  therefore  w'^  =  —  w^. 
In  this  example,  therefore,  the  values  of  the  function  in  the 
same  sheet  have  opposite  signs  on  the  opposite  sides  of  the 
branch-cut. 

Eiemann  calls  the  branch-points  also  windiTig-points,  because 
the  surface  winds  round  such  a  point  like  a  screw  surface  of 
infinitely  near  threads.  Then,  if  only  two  sheets  of  the  surface 
be  connected  at  such  a  point,  it  is  called  a  simple  hranch-point, 
or  a  winding-point  of  the  first  order ;  if,  however,  n  sheets  of 
the  surface  be  connected  at  it,  it  is  called  a  branch-jooint  or 
winding-point  of  the  (n  —  l)th  order.  Now,  for  many  inves- 
tigations, it  is  important  to  show  that  a  winding-point  of  the 
(n  —  l)th  order  may  be  regarded  as  one  at  which  n  —  1  simple 
branch-points  have  coincided.  If  we  assume,  for  example, 
n  =  5,  then  at  a  branch-point  in  which  5  sheets  are  connected, 
the  variable  passes  after  each  circuit  into  the  next  following 


70 


THEORY  OF  FUNCTIONS. 


sheet,  and  a  curve  must  make  5  circuits  round  a  branch-point 
before  it  arrives  again  in  the  first  and  becomes  closed.  By 
this  property  such  a  point  is  characterized.  But  the  same 
also  takes  place  if  we  assume  4  simple  branch-points  a,  b,  c,  d 
in   which  the  following  sheets  are  connected  in  succession: 


at 


a 
1  and  2 


b 
lands 


1  and  4 


d 
1  and  5. 


In  Fig.  16,  (m',  bb',  cc',  dd'  are  the  branch-cuts,  and  the  figures 
refer  to  the  numbers  of  the  sheets  in  which  the  lines  run. 
If  the  curve,  starting  from  z^  (Fig.  16),  cross  the  section  aa', 
it  passes  from  1  into  2  and  remains  in  2  for  the  entire  circuit, 

because  this  sheet  is  not  con- 
nected with  another  at  any  of 
the  points  6,  c,  d.  Thus,  for 
the  first  circuit  the  curve  passes 
from  1  into  2.  If  aa'  be  crossed 
a  second  time,  the  curve  passes 
from  2  into  1,  and  then  at  bb' 
from  1  into  3.  Then,  however, 
it  remains  in  3  until  its  return 
to  Zq,  the  second  circuit  there- 
fore carrying  it  into  3.  Only  at 
bb'  it  again  passes  from  3  into 
1,  and  then  at  cc'  from  1  into 
4.  In  this  way  each  new  cir- 
cuit carries  the  curve  into  the  next  following  sheet ;  therefore, 
after  the  fifth  circuit  the  curve  returns  into  the  first  sheet  and 
becomes  closed.  It  is  thus  seen  that  the  passages  take  place 
here  in  the  same  way  as  in  the  case  of  a  winding-point  of  the 
fourth  order.  Therefore,  by  making  the  four  simple  branch- 
points, as  well  as  the  branch-cuts,  approach  one  another  and 
finally  coincide,  we  obtain  a  branch-point  of  the  fourth  order. 
This  simple  example  shows  at  the  same  time  that  the  number 
of  circuits  which  a  curve  must  make  round  a  region  in  order  to 
become  closed,  exceeds  by  1  the  number  of  the  simple  branch- 
points contained  in  this  region,  since  the  winding-point  of  the 


Fig.  16. 


MULTIFORM  FUNCTIONS.  71 

fourth  order  is  equivalent  to  four  simple  branch-points.     It 
will  be  shown  later  that  this  relation  holds  generally. 

14.  For  the  complete  treatment  of  the  algebraic  functions 
it  is  still  requisite  for  us  to  take  into  consideration  infinitely 
large  values  of  the  variable  z.  In  the  plane  in  which  z  is 
moving,  this  variable  can  move  from  any  given  point,  e.g.,  from 
the  origin,  in  any  direction  to  infinity.  But  if  we  now  suppose 
the  plane  to  be  closed  at  infinity,  like  a  sphere  with  an  infi- 
nitely large  radius,  we  can  imagine  that  all  those  directions 
extending  to  infinity  meet  again  in  a  definite  point  of  the 
sphere,  and  accordingly  the  value  z  =  cc  can  then  be  repre- 
sented by  a  definite  point  on  the  spherical  surface.  The  same 
representation  may  be  obtained  by  conceiving  that  the  z-plane 
is  tangent  at  the  origin  to  a  sphere  of  arbitrary  radius.  Let 
us  suppose  the  point  of  tangency  to  be  the  north  pole  of  the 
sphere.  Then  any  point  z  of  the  plane  can  be  projected  on 
the  surface  of  the  sphere  by  drawing  a  straight  line  from  the 
south  pole  s  of  the  sphere  to  the  point  z,  and  cutting  with  this 
line  the  surface  of  the  sphere.  But  if  the  infinitely  distant 
points  of  the  z-plane  be  projected  in  this  manner  on  the  surface 
of  the  sphere,  the  projections  all  fall  in  the  point  s,  by  which 
point  therefore  the  value  2  =  co  is  represented  in  that  case. 

If,  now,  the  2;-plane  consist  of  ?i-sheets,  the  spherical  surface 
can  be  supposed  also  to  consist  of  ?i-sheets,  and  we  can  assume 
that  the  points  of  the  ?i-sheets  representing  the  value  z  =  00  lie 
directly  one  above  another.  Then  it  is  also  conceivable  that 
several  sheets  are  connected  at  the  point  00 ,  and  that  the  latter 
is  a  branch-point.  Given  a  function  w  =f(z),  in  order  to  decide 
whether  z  =  00   is   a  branch-point,   we   need  only   substitute 

z  =  —     If,  then,  f(z)  change  into  <f>  (u),  each  branch-point  z  =  a 
u 

of  f(z)  furnishes  for  <^  (u)  a  branch-point  u  =-,  and,  conversely, 

^  1 

each  branch-point  u=b  of  <fi (u)  furnishes  a  branch-point  2  =  - 

for  f(z) ;  therefore  z  =  00  is  or  is  not  a  branch-point  of  f(z), 
according  as  u  =  0  is  or  is  not  a  branch-point  of  <f>  (u).     We 


72  THEOBY  OF  FUNCTIONS. 

can  also  let  the  variable  z  describe  a  circuit  round  the  point  oo 
on  the  surface  of  the  sphere,  and  we  can  ascertain  whether  or 
not  f(z)  thereby  undergoes  a  change  in  value,  and  what  the 
nature  of  this  change  is,  if  for  this  purpose  we  examine  the 
function  <j>  (u),  while  the  variable  u  describes  a  circuit  round 
the  point  u  —  0. 

In  the  case  of  a  surface  closed  at  infinity,  a  branch-cut  can 
no  longer  be  drawn  extending  indefinitely  to  infinity,  but  if  such 
a  cut  extend  to  infinity,  it  now  terminates  in  the  definite  point 
z  =  00 .  Thence  arises  a  difficulty  which  has  to  be  removed 
(which,  however,  may  be  shown  to  be  only  apparent).  For, 
let  a,  b,  c,  •••  be  the  finite  branch-points  of  a  given  function 
f(z),  and  let  us  assume  that  these  points  are  connected  by 
means  of  branch-cuts  with  the  point  z  =  oo .  If,  now,  the 
values  of  the  function  Wi°,  w^,  •••,  w„°,  occurring  for  any  defi- 
nite value  Zq,  be  distributed  arbitrarily  among  the  points  z^, 
^ii  •••>  2:„°,  as  has  been  set  forth  on  page  65,  and  if  the  con- 
nection of  the  sheets  along  the  branch-cuts  be  determined  in 
accordance  with  the  character  of  the  function  f{z),  then  noth- 
ing arbitrary  must  be  assumed  for  the  point  z=  cxi ,  but  the 
manner  in  which  the  sheets  are  connected  along  the  branch- 
cuts  which  end  in  this  point  is  already  determined  by  the 
former  assumptions.  The  question,  however,  then  arises 
whether  this  actually  conforms  to  the  nature  of  the  function 
f{z),  i.e.,  whether  thereby  that  change  (or  eventually  non- 
change)  of  value  is  produced  which  f{z)  really  experiences  for 
a  circuit  round  the  point  z  =  oo  .  If  this  were  not  the  case,  it 
would  be  impossible  so  to  construct  the  Riemann  spherical 
surface  that  the  given  function  should  be  changed  into  a  one- 
valued  one  without  neglecting  the  infinitely  large  value  of  z. 

But,  now  in  the  simple  spherical  surface  a  closed  line  Z, 
which  encloses  the  point  z  =  oo  and  no  other  branch-point,  is 
at  the  same  time  one  which  encloses  all  finite  branch-points  a, 
b,  c,  •••.  Since  the  latter  can  be  resolved  into  closed  lines 
which  enclose  the  branch-points  a,  b,  c,  •••,  singly,  the  same 
change  of  values  takes  place  for  f(z)  in  describing  the  line  Z 
as  if  the  points  a,b,c,  •••  were  enclosed  singly  in  succession. 


MULTIFORM  FUNCTIONS.  73 

This,  however,  in  the  n-sheeted  surface,  as  has  been  shown 
above,  is  at  the  same  time  the  change  which  also  occurs  when 
the  branch-cuts  leading  from  a,b,c,  •••  to  infinity  are  crossed  in 
succession.  Accordingly,  there  arises,  in  fact,  no  contradiction, 
but  it  is  always  possible  to  represent  uniformly  an  algebraic 
function  by  a  many-sheeted  Kiemaim  spherical  surface,  without 
neglecting  the  infinitely  large  value  of  z. 

For  the  purpose  of  illustrating  the  above,  we  shall  introduce 
a  few  examples,  in  which,  for  the  sake  of  brevity,  the  same 
designations  will  be  used  as  have  been  employed  in  §  10  and 
§11. 

Ex.  1.   The  function  already  considered 


f(^)  =  yt 


b 


+  ^z  —  c 


changes  by  the  substitution  z  =  -  into 

u 


<^(^)=Vf 


au    vr 


bu  V^ 

hence  u  =  0,  and  therefore  also  z  =  oo,  is  a  branch-point,  and  it 
is  evident  that  at  this  point  the  same  sheets  are  connected  as 
at  the  point  c.  We  shall  therefore  draw  one  branch-cut  from 
a  to  b,  and  a  second  from  c  to  infinity.  But  it  is  also  possible 
to  draw  three  branch-cuts  from  a,  b,  c  to  infinity,  as  we  have 
done  in  the  general  treatment  of  this  subject.  Prom  the  con- 
siderations made  in  Ex.  4  of  §  10,  it  follows  that  the  passages 
along  the  branch-cuts  are  as  follows : 

123456 


along  a  oo 
«      6  CO 


231564 

123456 
312645 

«       123456 

"   COO'" 

456123 

If  therefore  for  a  single  circuit  round  the  point  oo  these  three 
cuts  be  crossed  successively,  we  then  pass  in  succession  first 


74 


THEORY  OF  FUNCTIONS. 


from  1  to  2,  then  to  1,  and  finally  to  4,  in  accordance  with  what 
should  really  occur. 


Ex.2. 


changes  into         <^  {u)  =  ^(i  _  an)  (1  -  hu)  \ 

therefore  2  =  oo  is  not  a  branch-point,  but  only  the  points  0,  a 

and  6. 

From  each  of  these  points  can  be  drawn  a  branch-cut  to 

infinity.     But  if  the  surface  be  assumed  to  be  closed  at  infinity, 

then  the  three  branch-cuts  meet 
at  the  point  oo  (Fig.  17).  The 
sheets  of  the  surface  pass  into 
one  another  along  the  part  a  oo 
in  a  way  different  from  that 
along  the  part  h  oo,  namely,  as 
the  numbers  indicate  in  the 
figure.  Round  the  branch- 
point 0,  in  the  direction  of 
increasing  angles,  /  iz)  changes 
into 


Fig.  it. 


AA? 


therefore  1  into  2,  and  hence  also  2  into  3  and  3  into  1.    If  we 

now  describe  a  circuit  round  the  point  oo,  then,  on  crossing  0  oo, 

1  changes   into  2,  and  on 

crossing  6  oo,  2  into  3,  and  /^  \  >^-~, 

finally  on  crossing  a  oo,  3      — 

intol.  Here  therefore  after  ^^»' 

the  first  circuit  we  return  to 

the  first  sheet,  the  function 

does  not  change  its  value, 

and  thus  the  point  oo  is 

really  not  a  branch-point. 

It  would  also  have  been  possible  in  this  case  to  connect  the 
points  a  and  6  by  a  branch-cut  drawn  in  the  finite  part  of  the 


V 


«  I 

y 


Fig.  is. 


MULTIFORM  FUNCTIONS. 


75 


surface  (Fig.  18).  But  then  there  must  be  given  on  this  line  a 
point  c,  at  which  separation  takes  place,  so  that  along  ac  the 
sheets  are  connected  in  a  different  way  from  that  along  be.  If 
then  the  second  branch-cut  be  drawn  from  0  to  c,  then  for  the 
circuit  round  the  point  c  the  function  remains  unchanged,  so 
that  c  is  not  a  branch-point.  The  matter  can  here  be  considered 
in  a  manner  analogous  to  the  treatment  of  the  second  example 
of  §  10,  namely,  as  if  the  point  c  had  arisen  from  the  coinci- 
dence of  three  branch-points  d,  e,f,  which  had  mutually  neu- 
tralized one  another,  so  that  the  given  function  may  be  supposed 
to  have  been  derived  from  the  following 


8/2  —  a    z  —  h     (z  —fY 
z~d    z—  e  z^ 


by  putting 


d  —  e—f=  c. 


The  branch-cuts  can  here  also  be  chosen  in  a  third  way  by 
drawing  one  from  a  to  0,  and  another  from  0  to  b. 

Ex.  3.   The  function  

f(z)  =  i/(z-a)iz~b) 
changes  into 


,  y  .        3/(1  —  au)  (1  —  bu) 

^('0=\^ ti ^; 


therefore  2;  =  oo  is  a  branch-point.    We  can  here  draw  a  branch- 
cut  from  a  to  00 ,  and  another  one  from  6  to  00  (Fig.  19),  and 
connect  the  sheets  as  indicated      o 
in  the  figure.     Then  a  circuit  i 
round  the  point  00  leads  first 
across  b  00  from  1  to  2,  and  then 
across  a  oo  from  2  to  3 ;  thus 
one  circuit  leads  from  1  to  3, 
so  that  the  function  changes 
and  z  =  ccis  actually  a  branch- 
point.    It  is  to  be  noted  that 
if    the    direction    of    motion 
here,  too,  be  that  of  increasing  fig.  19. 


76  THEORY  OF  FUNCTIONS. 

angles,  the  circuit,  viewed  from  oo ,  must  be  made  in  the  oppo- 
site direction.     For,  if  we  put 

u  =  r  (cos  <f>  —  i  sin  <^), 
it  follows  that 

z  =  -  (cos  d>  +  i  sin  ^). 
r 

Therefore,  if  u  describe  a  circle  round  the  origin  with  a  small 
radius  and  in  the  direction  of  decreasing  angles,  then  z  describes 

a  circle  with  a  large  radius  and 
/  \  in   the    direction   of   increasing 

3  12         ^'      angles.     In  this  case  <^(m),  for 


123 


V 


one   circuit,  changes  into  u-<f>  (u), 
2^  and  consequently  f(z)  into  aj'fiz); 

J  i.e.,  we  pass  from  1  to  3,  as  in- 


3 
2 
118 

dicated  in  the  figure. 


We  can  here  also  connect  the 

points  a  and  6  by  a  branch-cut 

pjg  20  running  in  the  finite  part  of  the 

plane,  assume  on  it  a  point  o* 

separation  c,  and  draw  from  this  another  branch-cut  to  co  (Fig. 

20).     The  function  then  does  not  change  for  a  circuit  round 

the  point  c. 

Ex.  4.  Let  us  next  take  up  the  example  previously  given 
on  page  38,  in  which  the  function  w  is  defined  by  the  cubic 
equation 

w^  —  w  -\-  z  =  0. 

By  letting  here  as  above 

3/ 


whereby  pq  =  ^,  (1) 

the  three  values  of  the  function  are  expressed  by 

W2=  ap-\-  a^q, 
w^  =  a^p  +  aq. 


MULTIFORM  FUNCTIONS.  77 

2 
We   have  here   first  the   two   branch-points   z  = and 

2  .  ^^ 

z  = ,  at  each  of  which  two  sheets  of  the  surface  are  con- 

V27 
nected ;  and  next  2  =  x  is  also  a  branch-point,  since  by  letting 

2  =  -,  we  get 
u 


3  \         27 

^  =  \ 2Z 


for  u=:0,  p  is  thus  =  00 ,  and  therefore  by  (1)  g  =  0.  Accord- 
ingly, at  2  =  00  all  three  values  of  the  function  become  infinitely 
large,  so  that  all  three  sheets  are  connected.  Although,  at  first 
sight,  it  seems  as  if  the  first  two  branch-points  must  have 
exactly  the  same  relation,  so  that  at  both  the  same  sheets  are 
connected,  yet  this  is  not  the  case.  To  see  this,  it  is  only  neces- 
sary to  follow  the  real  values  of  z,  since  the  first  two  branch- 
points are  real  and  the  point  z  =  00  can  also  be  assumed  on  the 
principal  axis ;  and  the  expressions  for  the  roots  w  must  be 
reduced  to  the  forms  which  are  given  to  them  in  the  irreduci- 
ble case  of  the  cubic  equation.     We  write  therefore 


and  let 

z  = 

V27 
Then  we  get 


2  o 

z  = cosS-y. 


p  =  -^ ^=  (cos S-y  +  I sinSv)  =  —  V^  (cosv  -f  isin-y) ; 

the  three  values  of  p  therefore  are 

p  =  -Vl  e",  =  -  a  Vi  e'",  =  -  a^*  V|  e", 

and,  since  q  is  always  =  — ,  the  corresponding  values  of  q  are 
3  J) 

g  =  -  VJ e-'",  =  -  aVJe^'",  =  -  a V|  e-'". 


78  THEORY  OF  FUNCTIONS. 

iirr  an- 

Since,  moreover,  a  =  e^ ,  a^  =  e    ^  can  be  substituted,  we  get 
wi  =  -  Vi  (e'"  +  e-'"), 

or  Wi  =  —  2  V^  cos  V, 

«;,  =  -2Vicos(^;  +  ^), 

W3  =  -  2 V|  cos  U ^j. 

As  long  as  v  remains  real,  z  passes  through  the  real  values 

2 
numerically  less  than  ;  the  point  z  therefore  describes  the 

V27 


distance   between    the    two    simple   branch-points;    for  pure 
imaginary  values  of  v,  z  becomes   numerically  greater  than 

2 

,  and  only  for  complex  values  of  v  does  z  assume  imaginary 


V27 

values.     For  our  purpose  therefore  only  the  real  values  of  v 

need  be  considered.  In  this  it  is  to  be  noted,  however,  that  z 
is  introduced  as  a  periodical  function  of  v.  Therefore,  if  we 
wish  to  have  something  definite  and  make  the  variable  z  de- 
scribe the  distance  between  the  two  simple  branch-points  only 
once,  we  must  choose  a  definite  period  and  consider  this  alone. 
Let  us  then  assume,  in  order  that  z  may  pass  through  the  real 

2               2 
values  -^ r=  to ^,  that  Sv  moves  from  0  to  tt,  and  there- 

V27  V72  . 

fore  V  from  0  to  -•     We  then  obtain  the  following  correspond- 

o 

ing  values  of  v,  z,  w^,  w^,  Wo : 


MULTIFOEM  FUNCTIOJSTS. 
Z  Wi  W2  Wf 

t 


+  -^        -2Vi        +Vl        +Vi 


V27 


-1  +1  0 


-vi     +2Vi     -vj 


In  calculating  them,  to  avoid  all  ambiguity  at  the  branch- 
points, we  must  start  from  the  value  v  =  -  and  make   it  first 

_  6 

decrease  to  0  and  then  increase  to  — 

^                       2  "^ 

If,  for  the  sake  of  brevity,  the  branch-points  -\ and  — 


^  V27 

be  denoted  by  e  and  e',  it  is  seen  that,  though  according  to  the 
chosen  period  of  v  the  two  values  of  the  function  Wj  and  Wg 
become  equal  at  e,  yet  after  z  has  arrived  at  e',  this  does  not 
again  take  place,  but  now  Wj  and  w^  become  equal.  Accord- 
ingly, at  e  the  sheets  2  and  3,  but  at  e'  the  sheets  1  and  3, 
must  be  assumed  to  be  connected. 

If,  according  to  the  above  table,  the  three  values  —  1,  4-1, 
0  of  10  occurring  for  z  =  0  be  distributed  consecutively  among 
the  sheets  1,  2,  3,  then,  for  a  circuit  round  the  point  e,  the 
values  -f- 1  and  0  interchange ;  while  for  a  circuit  round  the 
point  e'  the  values  —  1  and  0  interchange,  which  is  also  con- 
firmed by  a  direct  investigation  of  these  circuits.  Accordingly, 
if  the  branch-cuts  e  00  and  e'  oo  be  drawn,  the  continuation  of 
the  sheets,  in  crossing  them,  must  be  assumed  as  follows : 


along  e  oc 


123 
132 

123 
321* 


At  the  point  oo,  then,  all  three  sheets  are  connected,  into  which 
we  pass  successively  if  we  describe  a  circuit  round  this  point. 


80  THEORY  OF  FUNCTIONS. 

15.  If  w  denote  a  multiform  function  of  z,  but  W  a  rational 
function  of  w  and  z  (or  also  of  w  alone),  then  the  z-surface  for 
the  function  Wis  constructed  just  as  is  that  for  the  function  w. 
For,  let  w^^  and  w^  denote  any  two  values  of  w  belonging  to  the 
same  z,  and  W^  and  W^  the  corresponding  values  of  W,  then 
W^  must  change  into  W;^  Avhenever  w^  changes  into  w^,  since 
to  each  pair  of  values  of  z  and  iv  corresponds  only  a  single 
value  of  W.  The  passages  of  the  TT-values  depend  therefore 
upon  the  circuit  described  by  z  in  the  same  manner  as  do  the 
w-values. 

Therefore  the  z-surface  has  the  same  branch-points  and 
branch-cuts  for  W  as  for  w,  and  at  each  branch-point  the 
same  sheets  are  connected.  For  this  reason  Eiemann  calls 
all  rational  functions  of  w  and  z  a  system  of  like-branched 
functions. 


SECTION  IV. 

INTEGBALS    WITH   COMPLEX   VARIABLES. 

16.  The  definite  integral  of  a  function  of  a  complex  variable 
can  be  defined  in  exactly  the  same  manner  as  is  that  of  a 
function  of  a  real  variable. 

Let  Zq  and  z  be  any  two  complex  values  of  the  variable  z. 
Let  the  points  which  represent  these  values  be  connected  by 
an  arbitrary  continuous  line,  and  assume  on  it  a  series  of 
intermediate  points,  which  correspond  to  the  values  Zi,  Zj,  •••,z„ 
of  the  variable.  If,  further,  f(z)  be  a  function  of  z  which  at 
no  point  of  the  above  line  tends  towards  infinity,  and  if  we 
form  the  sum  of  the  products, 

/(2o)  (^l  -  ^o)  +f(Zl)  (Z2  -  Zl)  +  •••  +f(Zn)  (Z  -  2„), 

then  the  limit  of  this  expression,  when  the  number  of  the 
intermediate  values  between  %  and  z  along  the  arbitrary  line 


./ 


INTEGRALS   WITH  COMPLEX  VARIABLES.  81 

increases  indefinitely,  and  when  therefore  the  differences 
Zi  —  Zq,  Z2  —  Zi,  etc.,  diminish  indefinitely,  is  the  definite  integral 
between  the  limits  Zq  and  z ;  therefore 

(  f{z)  dz  =  lim  [/(zo)  (zi  -  z^)  +f(zi)  {z^  -  z,)  +  ••• 

+/(z„)(2-z„)].     (1) 

It  is  obvious  that  this  definition  does  not  essentially  differ 
from  that  usually  given  for  real  variables.  One  difference, 
however,  consists  in  this :  that,  in  accordance  with  the  nature 
of  a  complex  variable,  the  path  described  between  the  lower 
and  upper  limits,  i.e.,  the  series  of  intermediate  values,  is  not 
a  prescribed  one,  but  can  be  formed  by  means  of  any  continuous 
line.  Upon  the  nature  of  this  line,  which  is  called  the  path  of 
integration,  the  integral  is  in  general  absolutely  dependent. 
It  is  easy  to  show  that,  if  f(z)  do  not  become  infinite  at  any 
point  of  a  path  of  integration,  the  integral  taken  along  this 
/  path  has  also  a  finite  value.  For,  since  (§  2, 1)  the  modulus  of 
a  sum  is  Jess  than  the  sum  of  the  moduli  of  the  single  terms, 
it  follows  from  (1)  that 

mod  I    f{z)  dz  <  lim  \  mod  [/(zo)  (zi  —  Zq)] 

+  mod  [/(z,)  (z,  -  z,)]  +  . . .  +  mod  [/(z„)  (z  -  z„)]  ] . 

But  if  M  denote  the  greatest  of  the  values  acquired  by  the 
modulus  of  f(z),  while  z  describes  the  path  of  integration,  this 
value  according  to  the  assumption  being  finite,  then  the  right 
side  becomes  still  greater  if  M  be  put  in  place  of  the  moduli 
of  the  single  function-values  /(zo),  /(zi)---.  Therefore,  the 
modulus  of  a  product  being  equal  to  the  product  of  the  moduli 
of  the  factors  (§  2,  3),  we  get 

mod  I   f{z)  dz  <  M-  lim  {mod  (z^  —  Zq)  +  mod  {z^  —  z^)  +  ••• 

+  mod  (z  —  z„)|. 

In  this  the  moduli  of  the  differences  Zj  —  Zq,  Zj  —  Zi  •••  repre- 
sent the  lengths  of  the  chords  ZqZi,  z^Zj,  •••.     In  passing  to  the 


8^  THEORY  OF  FUNCTIONS. 

limit,  therefore,  the  sum  of  these  moduli  approaches  the  length 
L  of  the  path  of  integration ;  accordingly 


mod  Cf(z)dz<ML, 


and  has  a  finite  value,  if  the  path  of  integration  have  a  finite 
length.^ 

From  this  definition  follow  immediately  the  two  following 
propositions :  — 

1.  If  Zj  denote  any  value  of  the  variable  along  its  path,  then 

Cf{z)dz=  cy(z)dz+rf(z)dz. 

•^'o  ^*0  ^'k 

2.  Also  r V(2)  dz  =  -  f'fiz)  dz, 

i.e.,  if  the  variable  describe  the  path  which  represents  a  con- 
tinuous succession  of  its  values  in  the  opposite  direction,  the 
integral  assumes  the  opposite  sign. 

It  can  further  be  shown  that,  whatever  may  be  the  path  of 
integration,  the  integral 


is  always  a  function  of  the  upper  limit  z,  when  the  lower  limit 
Zq  is  assumed  to  be  constant.     Let 

Zo  =  Xo  +  iyo,  z  =  x  +  iy, 

I       f(i  +  ir])  (d$  +  idrj). 

This  integral  breaks  up  into  two  parts,  so  that  in  the  first  i, 
and  in  the  second  r],  is  the  variable  of  integration.  Given  now 
a  definite  path  of  integration,  then  by  virtue  of  it  t;  is  a  fimction 
of  i,  and  $  a  function  of  rj ;  let 

1  Konigsberger,  Vorlesungen  uber  die  Theorie  der  ellipt.  Funkt., 
I.  S.  63. 


INTEGRALS   WITH  COMPLEX  VARIABLES.  83 

If  these  be  introduced,  then,  since  $  passes  through  all  values 
from  Xq  to  X,  and  rj  at  the  same  time  through  all  values  from  yo 
to  y  corresponding  to  $  in  virtue  of  the  path  of  integration, 

^  =  rVC^  +  i<l>  (^)]  d4  +  i  f/C^  (77)  +  irf]  dr,,' 

and  this  equation  holds  whatever  may  be  the  functions  <f>  and 
\(/  determining  the  path  of  integration.  By  reducing  in  it  / 
also  to  the  form  of  a  complex  quantity,  we  are  led  to  only  real 
integrals,  and  hence  we  can  apply  to  the  former  the  rules 
of  differentiation  holding  for  real  integrals.  We  therefore 
obtain 

Y  =  (/■['A  (y)  +  %]  =  if(x  +  iy)- 

Hence  ^  =  i^; 

8y        8x' 

consequently  (by  §  5)  to  is  a  function  of  z.  It  then  follows, 
from  the  second  of  the  propositions  stated  above,  that  w  can 
also  be  considered  as  a  function  of  the  lower  limit  if  the 
upper   one   be   regarded   as   constant.     Since,   further   (§   5), 

dw  _  8w 
dz      8x 

it  follows  also  that  —  =  f(z). 

dz     -^^^ 

On  the  other  hand,  the  proposition  holding  for  real  integrals, 
that,  when  F(z)  denotes  a  function  of  z  the  derivative  of  which 
is  f(z), 

"^  f(z)dz  =  F{z)-F{z,), 


1  This  result  follows  also  from  the  sum  (1),  by  which  the  integral  is 
defined,  if  we  separate  in  it  the  complex  quantities  into  their  constituent 

parts. 


84  THEORY  OF  FUNCTIONS. 

cannot  without  further  limitation  be  applied  to  complex  in- 
tegrals, because  the  values  of  such  integrals,  as  has  already- 
been  remarked,  depend  not  only  upon  the  upper  and  lower 
limits,  but  also  upon  the  whole  series  of  intermediate  values, 
i.e.,  upon  the  path  of  integration. 

17.  In  order  to  examine  the  influence  of  the  path  of  integra- 
tion upon  the  value  of  the  integral,  we  shall  commence  with 
the  following  considerations.     Let 

z  =  x-\-  iy 

be  the  variable,  accordingly  x  and  y  the  rectangular  co-ordinates 
of  the  representing  point.  If  we  have  a  region  of  the  plane 
definitely  bounded  in  some  way,  which  can  consist  of  either 
one  or  several  sheets,  and  if  P  and  Q  be  two  real  functions 
of  X  and  y  which  for  all  points  within  the  region  are  finite  and 
continuous,  then  the  surface  integral 


-//(^-f)-- 


extended  over  the  whole  area  of  the  region,  is  equal  to  the  linear 
integral 

J(Pdx  +  Qdy), 

taken  round  the  whole  boundary  of  the  region. 

We  shall  not  only  prove  this  proposition  for  the  simplest 
case,  when  the  region  consists  of  only  one  sheet  and  is  bounded 
by  a  simple  closed  line,  but  we  shall  at  the  same  time  take 
into  consideration  those  cases  also  in  which  the  boundary  con- 
sists of  several  separate  closed  lines,  which  can  either  lie 
entirely  outside  of  one  another,  or  of  which  one  or  more  can 
be  entirely  enclosed  by  another.  Finally,  we  shall  not  exclude 
the  case  when  the  region  consists  of  several  sheets  which  are 
connected  with  one  another  along  the  branch-cuts.  Yet  we 
shall  then  assume  that  the  region  does  not  contain  any  branch- 
points at  which  the  functions  P  and  Q  become  infinite  or 
discontinuous.     It  is,  however,  necessary,  in  order  to  include 


INTEGRALS    WITH  COMPLEX   VARIABLES.  85 

all  those  cases,  to  determine  more  definitely  the  meaning  of 
houndary-direction.  If  we  assume,  as  is  customary,  that  the 
positive  directions  of  the  a>-  and  the  y-axis  lie  so  that  an 
observer  stationed  at  the  origin  and  looking  in  the  positive 
direction  of  the  x-axis  has  the  positive  ?/-axis  on  his  left,  then 
let  us  so  assume  the  positive  houndary-direction  that  one  who 
traces  it  in  this  direction  shall  always  have  the  bounded  area 
of  the  region  on  his  left.  The  same  can  be  expressed  thus: 
At  each  point  of  the  boundary  the  normal,  drawn  into  the 
interior  of  the  area,  is  situated  with  reference  to  the  positive 
direction  of  the  boundary  just  as 
is  the  positive  ?/-axis  with  refer- 
ence to  the  positive  a^axis.  If,  for 
instance,  the  boundary  consist  of 
an  external  closed  line  and  a  cir- 
cle lying  wholly  within  the  same, 
so  that  the  points  within  the  cir- 
cle are  external  to  the  bounded 
area  of  the  region,  then  on  the 
outer  line  the  positive  boundary- 
direction  is  that  of  increasing  an- 

1  1-1  1        •  •      1      •     •  Fio.  21. 

gles,  while  on  the  inner  circle  it  is 

the  opposite,  as  is  shown  by  the  arrows  in  Fig.  21.     Now  in 
the  linear  integral,  which  we  wish  to  prove  to  be  equal  to  the 
given  surface  integral,  the  integration  must  be  extended  over 
the  whole  boundary  in  the  positive  direction  as  just  defined. 
We  shall  write,  then,  the  integral  J  in  the  form 

and  we  can  then  integrate  in  the  first  part  as  to  x  and  in  the 
second  part  as  to  y.  For  this  purpose  we  divide  the  region 
into  elementary  strips,  which  are  formed  by  straight  lines 
lying  infinitely  near  to  one  another  and,  for  the  first  integral, 
running  parallel  to  the  ic-axis;  in  case  there  are  branch- 
points, we  take  care  to  draw  such  a  line  through  each  of  them. 
The  whole  region  is  thus  divided  into  infinitely  narrow  trape- 


86 


THEORY  OF  FUNCTIONS. 


Fig.  22. 


zoid-like  strips.     In  Fig.  22,  for  instance,  in  a  surface  consist- 
ing of  two  sheets  and  bovinded  by  a  closed  line  which  makes 

a  circuit  twice  round  a  branch- 
point, several  such  trapezoid- 
like  pieces  are  represented, 
the  lines  running  in  the  sec- 
ond sheet  being  dotted.  If 
we  now  select  some  one  of 
these  elementary  strips,  be- 
longing to  an  arbitrary  value 
of  y  (i.e.,  in  case  the  surface 
consists  of  several  sheets,  all 
those  elementary  strips  lying 
one  directly  below  another  in 
the  different  sheets  which  be- 
long to  the  same  value  of  y),  and  if  we  denote  the  values 
acquired  by  the  function  Q  at  those  places  where  the  ele- 
mentary strips  cut  the  boundary,  counting  from  left  to  right 
(i.e.,  in  the  direction  of  the  positive  avaxis),  at  the  points  of 
entrance  by 

Qi,  Qif  Qs,  •••, 

and  at  the  points  of  exit  by 

Q',  Q",  Q'",    -, 

then  (Fig.  23) 
therefore, 


1  It  must  be  noted  that  this  equation  remains  true,   even  when 


8x 


becomes  infinite  or  discontinuous  at  some  place  over  which  the  integration 
extends,  if  Q  suffer  no  interruption  of  continuity  at  this  place.  If, 
namely,  f(x)  be  a  function  of  the  real  variable  x,  which  f or  x  =  a  is 
continuous,  while  its  derivative,  /'(x),  is  for  the  same  value  discontinuous. 


INTEGRALS   WITH  COMPLEX  VARIABLES.  87 

In  the  integrals  on  the  right  y  passes  through  all  values 
from  the  least  to  the  greatest ;  therefore  dy  is  always  to  be 
taken  positively.  But  if  the  projections  on  the  y-axis  of  the 
elementary  arcs  which  have  been  cut  out  from  the  boundary 
by  the  elementary  strips  be  designated  in  the  same  sequence 
as  above,  at  the  places  of  entrance  by 

dyi,  dy^  dyg,  •••, 

and  at  the  places  of  exit  by 

dy',  dy",  dy'",  ..., 

we  assume  on  both  sides  of  a  two  values  Xh  and  Xt  infinitely  near  to  a. 
K,  then,  in  the  integral 


/ 


'^f(x)dx 


a  lie  between  the  limits  xo  and  xj,  and  if  /(x)  remain  continaoos  between 
the  same,  while  f(x')  is  discontinuous  only  at  the  place  x  =  a,  then 
we  can  put 

C''f(x)dx=  Urn  \  p*/(x)dz+   p/(x)dx1, 

wherein  the  limit  has  reference  to  the  coincidence  of  x»  and  x»  with  a. 
Now  since  /(x)  is  continuous  from  xo  to  x^  and  from  Xj  to  xi,  it  follows 
that 

Cyix)dx  =  Um  [/(x*)  -  /(Xo)  +  /(xi)  -  /(xt)]. 

Since /(x)  is  continuous  at  the  place  x  =  a,  therefore,  in  passing  to  the 
limit, /(x»)  and/Cxi)  become  equal,  or 

lim[/(x»)-/(a:*)]=0; 

therefore,  notwithstanding  the  discontinuity  of  f(x)  between  the  limits 
of  the  integral,  we  have 


c/x„ 


y(x)(Zx=/(xi)-/(xo). 


This  case  deserves  notice  here,  since  it  will  be  shown  later  that  the 
derivatives  of  continuous  functions  can  become  infinite  at  branch-points 
(§39). 


88 


THEORY  OF  FUNCTIONS. 


and  if  regard  be   paid  to  the   positive    boundary   direction 
(Fig  23),  then 

dy  =  —  dyi  =  -  dy^  =  -  dyg  =  ... 

=  +  dy'  =  +  dy"  =  +  dy"^=.:; 
therefore, 


jP^^^^y  =fQidyi  +fQ'dy'  +fQ^y, 


+ 


Fig.  28. 


In  all  these  integrals  y  changes  in  the  setise  of  the  positive 
boundary- direction ;  therefore  they  all  reduce  to  a  single  one, 
and  we  have 


/J|^^^%=Jq%, 


if  the  latter  integral  be  extended  along  the  entire  boundary  in 
the  positive  direction. 

In  the  same  manner  the  second  integral 


fn*^'' 


can  be  treated.  Here  the  region  is  divided  into  elementary 
strips  by  straight  lines  running  parallel  to  the  y-axis,  and,  as 
before,  such  a  line  is  drawn  through  each  branch-point.  If, 
therefore,  the  values  which  the  function  P  has  at  the  places 


INTEGRALS   WITH  COMPLEX  VARIABLES.  89 

where  an  elementary  strip  cuts  the  boundary  be  designated,  in 
order  from  below  upward  {i.e.,  in  the  direction  of  the  positive 
y-axis),  at  the  places  of  entrance  by 

and  at  the  places  of  exit  by 

pi   pii   pill    .. 
then  again 

r  C^dxdy  =  -  CPjdx  +  fP'dx  -  Cp^dx  +  •  •  • ; 

and  therein  dx  is  positive.     But  if 

dx^  dx2,  dx^,  ••-,  and  dx',  dx",  dx'",  ••• 

designate  the  projections  of  the  elementary  arcs  which  are  cut 
out  by  the  elementary  strips,  then,  considering  the  positive 
direction  of  the  boundary, 

dx  =  -\-  dxi  =  4-  dx2  =  +  dxg  =  ••  • 

=.-dx'  =  -dx"  =  -dx'"="., 
and  therefore 

C  C^dxdy  =  -  CPidx^  -  Cp'dx'  -  C P^dx^ , 

=  -  fPdx, 

in  which  the  integral  is  to  be  extended  in  the  positive  direction 
round  the  entire  boundary.  Combining  the  two  integrals,  it 
follows,  as  was  to  be  proved,  that 

the  linear  integral  to  be  taken  round  the  entire  boundary  in  the 
positive  direction. 

This  proposition,  which  is  hereby  proved  for  the  real  func- 
tions P  and  Q,  can  at  once  be  extended  to  the  case  when  P  and 
Q  are  complex  functions  of  the  real  variables  x  and  y.     If  we 

^""^  P^P'  +  iP",  Q=Q>  +  iQ", 


90  THEORY  OF  FUNCTIONS. 

wherein  P',  P",  Q',  Q"  are  real  functions  of  x  and  y,  then 

If  the  proposition  be  applied  to  the  right  side  of  the  equation, 
we  get 

=  j'(P'dx  +  Q'dy)  +  ifi^'dx  +  Q"dy)  =j'(Pdx  +  Qdy). 

We  have  assumed  until  now  that,  within  the  region  under 
consideration,  there  are  no  branch-points  or  other  points  at 
which  P  and  Q  are  discontinuous.  Now,  in  order  to  include 
within  our  considerations  also  those  regions  in  which  this  is 
the  case,  it  is  only  necessary  to  enclose,  and  thereby  exclude, 
such  points  by  arbitrary  small  closed  lines,  these  new  lines 
then  forming  part  of  the  boundary  of  the  region. 

18.  From  the  preceding  proposition,  follows  immediately 
the  following:  — 

(i.)  If  Pdx  +  Qdy  be  a  complete  differential,  then  the  integral, 

I  (Pdx  +  Qdy),  extended  over  the  whole  boundary  of  a  region 

within  which  P  and  Q  are  finite  and  continuous,  is  equal  to  zero. 
For,  if  Pdx  +  Qdy  be  a  complete  differential, 

hy      8x 

and  therefore  all  the  elements  of  the  surface  integral,  which  is 
equal  to  the  linear  integral,  disappear,  and  accordingly  this,  as 
well  as  that,  is  equal  to  zero. 

If  now  w=f(z) 

be  a  function  of  a  complex  variable  z  =  x  +  iy,  then  [§  5.  (1)] 

Sw_  .ho  _8(iw) 
'8y~^Sx~    8a;  ' 


INTEGRALS    WITH  COMPLEX  VARIABLES, 


91 


therefore         wdx  +  iwdy,  i.e.,  w  (dx  +  idy),  or  wdz 
is  a  complete  differential,  and  hence 

(ii.)  I  f(z)  dz  =  0,  if  this  integral  be  extended  round  the  whole 
boundary  of  a  region  within  which  f(z)  isjtnite  and  continuous. 

From  this  follows  further:  If  the  variable  z  be  made  to 
describe  between  the  points  a  and  b  two  different  paths  acb 
and  adb  (Fig.  24),  forming  together  a  closed  j, 

line  which  in  itself  alone  is  the  complete  boun- 
dary of  a  region,  and  if  f(z)  be  finite  and  con- 
tinuous within  this  region,  then,  for  the  integral 
extended  round  the  closed  line,  we  have 


Jf(z)dz 


0. 


Fig.  24. 


In  order  to  designate  briefly  an  integral  taken 
along  a  definite  path,  we  shall  choose  the  letter 
J"  and  add  to  it  the  path  of  integration  in  pa- 
renthesis,  so  that,  for   instance,   the   integral 

if(z)dz,   taken   along  the  path   acb,  will   be  designated  by 

J  (acb).     The  last  equation  can  be  written 

J(acbda)  =  0. 

But  (§  16)        J{acbda)  =  J(acb)  +  J(bda) 

and  J(bda)  =  —  J  (adb) ; 

it  follows,  therefore,  that  J  (acb)  =  J  (adb). 

(iii.)    The  integral  |  f(z)dz,  therefore,  has  always  the  same 

value  along  two  different  paths  joining  the  same  points,  if  the  two 
paths  taken  together  be  the  boundary  of  a  region  in  which  f(z)  is 
finite  and  continuous. 

If  we  have  a  connected  region  within  which  f(z)  remains 
finite  and  continuous,  of  such  a  nature  that  every  closed  line 
described  in  it  forms  by  itself  alone  a  complete  boundary  of  a 


92 


THEORY  OF  FUNCTIONS. 


part  of  the  region,  then  the  integral    |  f{z)  dz  has,  along  all 

paths  between  the  two  points,  the  same  value.  Let  the  lower 
limit  Za  be  constant ;  then  within  such  a  part  of  the  region  the 
integral  is  a  uniform  function  of  the  upper  limit,  and  if  F(z) 
denote  a  function  the  derivative  of  which  is  f{z),  then  within 
this  region 

"y{z)dz  =  F{z)-F(zo), 


f 


since  in  this  case  the  value  of  the  integral  is  independent  of 
the  path  of  integration.  The  great  importance  of  those  sur- 
faces, in  which  each  closed  line  forms  by  itself  alone  the  com- 
plete boundary  of  a  region,  becomes  here  quite  evident. 

Riemann  has  called  surfaces  of  such  a  character  simply 
connected  surfaces.  Such  is,  for  instance,  the  surface  within  a 
circle.  If  f(z)  be  continuous  everywhere  in  such  a  surface, 
then,  as  has  been  noted, 


Cf{z)dz 


is  a  uniform  function  of  the  upper  limit.     If,  on  the  other 
hajid,/(z)  become  infinite  within  the  surface  of  a  circle,  for 

instance,  only  at  one  point  a,  and 
if,  in  order  to  obtain  a  part  of  the 
region  within  which  f{z)  remains 
continuous,  a  small  circle  fc  be  de- 
scribed round  this  point,  thereby 
excluding  it,  then  the  ring-shaped 
portion  of  the  plane  thus  obtained 
is  no  longer  simply  connected;  for 
a  line  w,  which  encloses  entirely 
the  small  circle,  does  not  form  by 
itself  alone  the  entire  boundary  of 
a  part  of  the  region,  but  only  m  and  fc  together.  Accordingly 
the  integral  extended  along  m  and  h  together  has  the  value 
zero ;  but  if  the  integral  extended  along  Zc  alone  be  not  equal 
to  zero,  the  integral  taken  along  m  cannot  be  zero.  Within 
such  a  region,  which  Riemann  has  called  multiply  connected, 


Fig.  25. 


INTEGRALS   WITH  COMPLEX  VARIABLES. 


93 


the  dependence  of  the  integral  upon  the  path  of  integration 
continues,  and  the  integral  can  be  regarded  as  a  multiform 
function  of  the  upper  limit.^ 

19.  We  now  drop  the  assumption  that  the  function  f(z)  in 
the  region  under  consideration  is  everywhere  continuous,  and 
we  proceed  to  investigate  those  integrals  of  which  the  paths  of 
integration  are  boundaries  of  regions  in  which  the  function  is 
not  everywhere  continuous.  If  f(z)  be  infinite  or  discontinuous 
at  any  point  of  a  region,  then  such  a  point  is  to  be  called  a 
point  of  discontinuity.  It  may  or  may  not  be  at  the  same  time 
a  branch-point.  If  there  be  points  of  discontinuity  in  a  region 
of  a  plane,  we  are  no  longer  justified  in  all  cases  in  concluding 
that  the  integral,  extended  over  the  whole  boundary  of  the 
region,  has  the  value  zero,  because  the  proof  of  this  proposition 
rests  essentially  upon  the  assumption  that/(2:)  does  not  become 
discontinuous  within  the  region.  But  the  following  can  be 
proved : — 

(iv.)  Whatever  may  be  the  value  of  the  integral,  it  does  not 
change  if  the  region  be  increased  or  diminished  by  arbitrary  pieces, 
provided  only  that  f(z) 
is  finite  and  continuous 
within  the  added  or  sub- 
tracted jneces.  For,  if  in 
the  first  place  an  added 
or  subtracted  piece  be 
completely  bounded  by 
one  line,  as,  for  in- 
stance, A  OT  B  (Fig.  26, 
where  abcda  is  the  orig- 
inal boundary),  then,  if 
f(z)  be  continuous  within 
A  and  B,  the  integral  extended  over  the  boundary  of  -4  or  £ 
must  be  zero.  The  boundary  of  ^  or  5  can  therefore  be  arbi- 
trarily added  to  the  original  one  without  changing  the  value 


Fig.  26. 


1  See  Sections  IX.  and  X. 


94  THEORY  OF  FUNCTIONS. 

of  the  integral.     If,  however,  the  added  or  subtracted  piece 

be  bounded  in  part  by  the  original  contour,  as  hfdcb  or  bcdeb, 

then 

J(bfd)  =  J  (bed)  =  J  (bed), 

iif(z)  be  continuous  within  these  regions.  Therefore  the  por- 
tion of  the  boundary  bed  can  be  replaced  arbitrarily  by  either 
bfd  or  bed  without  altering  the  value  of  the  integral.  From 
this  it  follows  further  that  a  closed  line,  which  either  forms 
alone  the  boundary  of  a  region  or  at  least  forms  part  of  such  a 
boundary,  can  also  be  replaced  arbitrarily  by  a  more  extended 
or  contracted  closed  line,  provided  only  that  no  portions  of  the 
surface  are  thereby  either  added  or  subtracted  in  which  f(z) 
becomes  infinite  or  discontinuous.  For,  in  order  to  extend,  for 
instance,  abcda  into  ghkg,  it  is  only  necessary  to  replace  first 
bed  by  bhkd,  and  then  Jcdabh  by  kgh.  In  a  similar  manner  the 
validity  of  the  proposition  can  be  proved  in  all  cases.  Its 
general  application,  however,  even  to  regions  which  consist  of 
several  sheets  or  contain  gaps,  can  be  demonstrated  in  the 
following  way.  If  an  arbitrary  surface  T  be  so  divided  into 
two  parts,  M  and  N,  that  f(z)  is  continuous   in   31,  and   if 

the  integral   I  f(z)  dz,  extended  over  the  boundary  of  one  part, 

say  M,  be  designated  by  J(M),  then 

J(M)  =  0. 

If,  now,  the  portions  M  and  N  have  no  common  boundary- 
pieces,  the  boundaries  of  M  and  N  together  form  the  boundary 
of  T,  and  therefore 

J(T)=J(M)  +  J(N); 

consequently  also  J(T)  =  J(^)- 

If,  however,  certain  lines  G  form  part  of  the  boundaries 
of  both  M  and  iV,  then  the  pieces  M  and  N  lie  on  opposite 
sides  of  this  line  C.  If,  therefore,  the  boundaries  3f  and  JSf  be 
described  successively  in  the  positive  boundary-direction,  i.e., 
so  that  the  bounded  region  is  always  on  the  left,  then  the  lines 


INTEGRALS   WITH  COMPLEX  VARIABLES.  95 

C  are  described  twice,  in  opposite  directions;  consequently 
the  integrals  extended  along  C  cancel  each  other,  while  the 
remaining  boundary -pieces  of  M  and  N  form  the  entire  boun- 
dary of  T;  therefore 

J(T)  =  J(3q+J{N), 

and  consequently  J{T)  =  J{N). 

Now,  just  as,  according  to  this,  the  part  M  can  be  sepa- 
rated from  the  surface  T,  so,  conversely,  a  surface  N  can 
be  extended  by  the  addition  of  a  surface  M  in  which  the 
function  remains  continuous,  without  changing  the  boundary- 
integral. 

From  this  another  important  proposition  can  be  deduced.  If 
a  closed  line  (/)  form  by  itself  alone  the  complete  boundary 
of  a  region,  and  if  the  function  f(z)  become  discontinuous 
within  it  at  the  points  aj,  a^,  Ug,  •••,  let  each  one  of  these  points 
be  enclosed  by  an  arbitrary  small  closed  line,  say  by  a  small 
circle,  which,  however,  in  case  one  of  these  points  of  discon- 
tinuity be  at  the  same  time  a  branch-point,  must  be  described 
as  many  times  as  there  are  sheets 
connected  at  it;  then  all  these  cir- 
cles, which  may  be  designated  by 
(^i),  (^2),  (A),  ■■;  form,  together 
with  the  outer  line  (/),  the  boun- 
dary of  a  region  in  which  f(z)  is 
continuous  (Fig.  27,  in  which  the 
dotted  lines  run  in  the  second  sheet). 

Consequently  the  integral    I  f(z)  dz, 

extended  in  the  positive   direction 

over  the  whole  boundary,  is  equal  to  zero.  But  if  the  outer 
line  (/)  be  described  in  the  direction  of  increasing  angles, 
the  small  circles  (^1),  (^2)5  (-^s),  •••  must  be  described  in 
the  direction  of  decreasing  angles.     If,  therefore,  the  integral 

f(z)dz,  extended  in  the  direction  of  the  increasing  angles 


s 


96  THEORY  OF  FUNCTIONS. 

along  the  lines  (/),  (^i),  (A2),  (A3),  •••,  be  designated  by  /,  Ai^ 
A2,  A3,  •••,  then 

J  —  Ai  —  A2  —  A3  —  •  •  •  =  0, 

and  consequently 

1=  Ai  +  A2  +  A3-\-  ■■: 

If  now  the  line  (I)  be  described  in  a  region  T,  which 
contains  no  other  points  of  discontinuity  than  the  above 
ai,  tto,  ttg,  •••,  then  the  integral  /,  according  to  the  last  proposi- 
tion, retains  its  value  if  it  be  extended  over  the  boundary  of 
T;  we  thus  obtain  the  proposition :  — 

(v.)    The  integral   i  f{z)  dz,  extended  over  the  whole  boundary 

of  a  region  T,  is  equal  to  the  sum  of  the  integrals  along  small  closed 
lines  which  enclose  singly  all  the  points  of  discontinuity  contained 
within  T,  all  the  integrals  being  taken  in  the  same  direction. 

20.  By  the  preceding  considerations  we  are  led  to  the  inves- 
tigation of  such  closed  paths  of  integration  as  enclose  only 
one  point  of  discontinuity.  We  must  distinguish,  however, 
whether  the  point  of  discontinuity  is  or  is  not  at  the  same 
time  a  branch-point.  Let  us  consider  first  a  point  a,  which  is 
not  a  branch-point,  and  at  which  f(z)  becomes  infinite.  If  the 
integral 

A=j'f(z)dz 

be  taken  along  a  line  enclosing  one  of  the  points  of  discon- 
tinuity, this  line  enclosing  neither  another  point  of  disconti- 
nuity nor  a  branch-point,  then  the  path  of  integration  can  be 
replaced  by  a  small  circle  described  round  the  point  a  with  the 
radius  r,  which  can  be  made  to  tend  towards  zero  without 
changing  the  value  of  the  integral.     If  we  write 

A=  f{z-a)f(z)-^, 
J  z  —  a 

and  let  z  —  a  =  r  (cos  <^  +  i  sin  </>), 


INTEGRALS   WITH  COMPLEX  VARIABLES.  97 

tlien  r  remains  constant,  and  <f>  increases  from  0  to  2  tt  when  z 
describes  the  small  circle.  Here  it  is  assumed  that  the  point 
starts  from  the  point  Zq,  at  which 
the  line  drawn  through  a  in  the  posi- 
tive direction  parallel  to  the  princi- 
pal axis  cuts  the  circle  (Fig.  28). 
This  is  permissible,  since  the  ini- 
tial point  of  the  description  can  be 
chosen  arbitrarily.     Now,  in  order     , 

to  express  in  terms  of  z  and  fig  28 

z  —  a 

<fi,  we  remark  with  Riemann,  that  dz  denotes  an  infinitely 
small  arc  of  a  circle,  starting  from  any  point  on  the  circum- 
ference and  subtending  the  angle  d<f>  at  the  centre.  If  the 
terminal  point  of  this  infinitely  small  arc  of  the  circle  be 
designated  by  z',  then 

,         ,  dz        z'  —  z 

dz  =  z'  —  z,      = 

z  —  a     z  —  a 

But  in  §  2,  page  21,  it  has  been  shown  that 

=  =  (—  cos  a  +  ^  sm  a), 

z  —  a      az 

wherein  a  is  the  angle  azz',  in  this  case  a  right  angle ;  therefore 

dz    _  .zz' 


z  —  a 


The  line  zz'  is  an  arc  of  a  circle  with  the  angle  d<f>  at  the 

centre,  therefore  it  equals  rd<t>,  and  az  is  equal  to  the  radius  r ; 

accordingly  we  get 

dz         .rd<i>      ...  I 

z  —  a  r 

1  From  z  —  a  =  7-  (cos  <f>  +  i  sin  (p) 

we  also  get  by  direct  differentiation,  tlie  radius  r  remaining  constant, 

dz  =  r  (—  sixKp  +  I  cos  0)  d4> 

=  ir  (cos  0  +  I  sin  <p)  d<p ; 

dz 
therefore  =  id<f>. 


98  THEORY  OF  FUNCTIONS. 

If  this  result  be  substituted  in  the  integral,  it  follows  that 

o  A=  \     (z  —  a)f{z)  id<t}. 

If  the  radius  r  be  made  to  decrease  indefinitely,  the  points  z 

of  the  circumference  of  the  circle  approach  the  point  a;  z  —  a 

*  therefore  approaches  zero,  while  f(z)  becomes  infinite.     If  it 

U^  happen  that  f{z)  becomes  finite  for  z  =  a  in.  such  a  way  that 

the  product  (z  —  a)f{z)  tends  towards  a  definite  finite  limit 

p,  i.e.,  if 

lim[(z-a)f(z)'],^,=  p, 

wherein  it  is  expressly  assumed  that  this  limiting  value 
always  remains  the  same  from  whatever  side  the  point  z  may 
approach  the  point  a,  then  we  can  assume  that,  for  all  points  z 
in  the  vicinity  of  the  point  a, 

(z-a)f(z)=p  +  e, 

wherein  c  denotes  a  function  of  r  and  <j>  which  becomes  infini- 
tesimal with  r  for  any  value  of  <f>.     Then 


idcf)  -f  I     ddif). 

0  *^0 


If  r,  and  therefore  also  c,  be  made  to  vanish,  then  the  second 
integral  also  vanishes,  and  it  follows  that 

A  =  2  Trip. 

The  value  of  the  integral  is  thereby  expressed  in  terms  of 
the  limiting  value  of  (z  —  a)f(z),  when  this  is  finite  and 
determinate.  This  value  of  A,  by  (iv.),  does  not  change  if  the 
integration  be  extended  over  the  complete  boundary  of  a  region 
within  which  there  are  no  points  of  discontinuity  except  a. 

The  integral  f    ^^ 

Jl  +  z' 

may  serve  as  an  example. 
Here  f{z)  = 


1  +  z'  (z-i)(z+iy 


INTEGRALS   WITH  COMPLEX  VARIABLES. 


99 


which  becomes  infinite  for  z  =  i,  the  point  2  =  t  not  being  a 
branch-point  (the  function has  no  branch-points  what- 


ever). 
Further, 


l+z^ 


^=^'"[rTil./ 


lim 


z  -\-i 


consequently 


the  integral  being  extended  over  a  line  enclosing  the  point  i  in 
the  direction  of  increasing  angles. 

If  in  a  region  T  there  be  the  points  of  discontinuity  0^,  a^, 
Os,  •••,  which  cannot  at  the  same  time  be  branch-points,  and  if 
/(z)  become  infinite  at  these  points  in  such  a  way  that  the 
products  (z  — ai)/(z),  {z  —  a^f{z),"'  approach  determinate 
finite  limiting  values  pi,  j92>  •••>  i-^-i  if 

lim  [(z  -  ai)/(z)]^^  =i9i, 

lim  [(z  -  a2)/(z)],.    =  p^ 


then  the  integral   i  /(z)  dz,  extended  over  the  whole  boundary 
of  T,  assumes  the  value  [(v.),  §  19]. 

\  f(z)dz  =  2  Trt (i?i  +P2  +i?3  +  •••)• 

In  the  preceding  example 

is  infinite  also  for  z  =  —  i,  and  for  this  point  we  get 


therefore,  also, 


dz 


'    1    ' 
z  —  i 


2i' 


h 


-\-z' 


taken  along  a  line  enclosing  —  i. 

For  a  line  enclosing  both  points  +  i  and  —  i  in  the  direction 
of  increasing  angles,  this  integral  becomes  tt  —  tt  =  0. 


100 


THEORY  OF  FUNCTIONS. 


Now  by  means  of  such  closed  lines  as  include  only  a  single 
point  of  discontinuity  it  is  possible,  within  a  region  containing 
no  branch-points  nor  any  gaps,  to  refer  to  one  another  the 
values  of  the  integrals  for  the  different  paths  of  integration. 
If  two  paths  bee  and  bdc  (Fig.  29)  enclose  only  one  point  of 
discontinuity  a  and  no  branch-point,^  then  the  one,  say  bdc, 
can  be  replaced,  by  enclosing  the  point  of  discontinuity  in  a 
closed  line  bghb  before  describing  the  other  path  bee.  Then 
by  (iv.),  §  19, 

J(bghb)  =  J(bdceb)  =  J(bdc)  -  J{bec), 
therefore  J  (bdc)  =  J(bghb)  +  J{bec), 

or  also  J  (bee)  =  —  J  (bghb)  -\-  J  (bdc) 

=  J{bhgb)  +  J(bdc). 


Fig.  29.  Fig.  30. 

We  get  a  similar  result  if  two  paths  enclose  several  points 
of  discontinuity,  but  no  branch-points.  For  instance,  let  the 
paths  Zffid  and  z^ed  (Fig.  30)  enclose  two  points  of  discon- 
tinuity a  and  b,  and  draw  from  Zq  round  each  of  them  a  closed 
line  ZfifgzQ  and  z^fikz^.     Then 

J(zohkZo)  +  J(zofgzo)  =  J(zoedcZo) 

=  J(Zoed)  —  J(zocd)  ; 
consequently 

J{Z(fid)  =  J(zofgzo)  +  J(Zffi,kzo)  +  J{Z(fid). 

1  The  assumption  that  the  two  paths  enclose  no  branch-point  is,  in 
general,  necessary  only  in  order  that  together  they  may  form  a  complete 
boundary,  which  may  not  always  be  the  case  if  there  be  branch-points 
between  them. 


INTEGRALS   WITH  COMPLEX  VARIABLES.         101 

Therefore  the  one  path  can  be  replaced,  by  describing  closed 
circuits  round  each  of  the  points  of  discontinuity  before  describ- 
ing the  other  path. 

The  properties  of  the  integral  A  round  a  point  of  discon- 
tinuity, if  {z  —  a)f(z)  no  longer  have  a  determinate  finite 
limiting  value  at  that  point,  cannot  be  discussed  until  later 
(Section  VIII.). 

21.  We  next  proceed  to  the  case  when  the  point  of  discon- 
tinuity is  at  the  same  time  a  branch-point,  in  which  case  it 
will  be  denoted  by  6,  and  the  value  of  the  integral,  for  a  line 
described  round  it,  by  B.  We  assume  that  at  this  point  m 
sheets  of  the  surface  are  connected.  If  we  wish  to  have  here 
a  line  enclosing  the  point  h,  it  must  make  m  circuits  round  h ; 
e.g.,  let  it  describe  the  circumference  of  a  circle  m  times. 
Riemann  introduces  in  this  case,  in  the  place  of  z,  a  new 
variable  t,,  letting 

which  therefore  receives  the  value  0  for  z  =  h;  and  he  inquires 

how  the  function  f(z),  considered  as  a  function  of  ^,  behaves 

at  the  point  ^  =  0.     For  this  purpose  we  first  determine  what 

line  is  described  by  ^  when  z  describes  a  closed  circle,  i.e.,      )   "^  "^ 

makes  uti  circuits  round  the  latter.  1  ^  [r 

If  we  let  z  —  h  =  r  (cos  9  +  i  sin  6), 

and  therefore         ^  => »^ (cos  —  ^  +  i sin i ^  ), 
\       m  m  j 

1 
then  r,  and  consequently  also  )•",  remains  constant,  and  there- 
fore ^  also  describes  a  circle,  namely,  one  round  the  origin. 
But  after  z  has  completed  one  circuit,  so  that  6  has  increased 

1  2 

from  0  to  2tr,  then  —6  has  increased  from  0  to  —;  conse- 
m  m 

quently  ^  has  described  the  7)ith  part  of  the  circumference. 

For  the  second  circuit  of  z,  ^  again  describes  the  ?n,th  part  of 

the  circumference,  and  likewise  for  each  new  circuit  of  z. 


102 


THEORY  OF  FUNCTIONS. 


Fig.  81. 


Consequently,  after  z  has  made  m  circuits  and  returned  to  its 
starting-point,  ^  has  described  the  entire  circumference  of  the 

circle  exactly  once.  Therefore, 
to  the  m  pieces  of  the  region 
covered  by  the  radius  r  during 
these  circuits,  correspond  m  sec- 
tors of  the  circle,  each  with  the 

angle  —  at  the  centre.  These  ioin 

one  another  and  form  together  a 
simple  circular  surface.  In  Fig. 
31,  it  is  assumed  that  at  the  point 
b  three  sheets  are  connected, 
which  continue  into  one  another 
along  the  branch-cut  bb'.  The 
circular  lines  running  in  the  three 
sheets  have  been  drawn  for  the 
sake  of  clearness  side  by  side, 
and  the  lines  running  in  the  1st,  2d,  and  3d  sheets  are  repre- 
sented by  a  continuous  line,  a  thickly  dotted  and  a  thinly 
dotted  line,  respectively.     Then 

to  the  surface  cd^orresponds  the  sector  of  the  circle  c'oe', 

II      u  a         Qfg  a  u         u         «      u        u       gf^gi^ 

"    "        "       gh/        "  "       "       "    "      "     g'oc', 

and  therefore  to  the  whole  area  of  the  surface  bounded  by  the 
closed  line  cdefghc  corresponds  the  simple  circular  surface 
c'e'g'c'.  It  follows  therefore  that,  while  z,  passing  through  all 
the  m  sheets,  returns  to  its  starting-point  only  after  m  circuits, 
I  does  so  after  the  first  circuit.  The  variable  ^  therefore  does 
not  leave  its  first  sheet,  and  consequently  the  function  f(z), 
considered  as  a  function  of  ^,  does  not  have  a  branch-point  at  the 
place  ^=0.    Accordingly,  if  ^  be  introduced  as  the  variable  in  the 

integral  |  f(z)  dz,  extended  over  a  circuit  enclosing  the  branch- 
point and  rh^  point  of  discontinuity  b,  the  considerations  of 
the  last  paragraph  can  be   applied,  because   ^  =  0  is  not  a 


INTEGRALS   WITH  COMPLEX  VARIABLES.         103 

branch-point,  but  merely  a  point   of   discontinuity.     Making 

the  substitution  ^=  (z  —  6)",  suppose  f{z)  changes  into  <l>  (^)  ; 
then,  since  dz  =  m^  ~  ^  dt,, 

If,  for  the  sake  of  brevity,  we  put  —  6  =  il/,  and  therefore 

m 

^  =  r^  (cos  f  +  i  sin  i/r), 
then  if/  increases  for   the  whole  circuit  from   0  to  2  7r;    and, 
since  as  above  —  =  idij/,  it  follows  that 


B  =  m,r''tr<j>{C)id^. 


According  to  the  assumption  <^  (^)  is  infinite  for  ^  =  0.  But 
if  the  tendency  to  become  infinite  be  of  such  a  nature  that  one 
of  the  products 

approaches  a  finite  limiting  value,  then 
lim[r<^(0]?=.o  =  0. 

Therefore,  if  the  radius  r  of  the  circle  described  round  the 
/-f  ^  point  b  tend  towards  0,/then  ^  =  0.  Illlllxt'fl^ 

y^^~    J         If  we  now  return  to  the  variable  z,  we  obtain  the  proposi-    ^  tjty^.Mj     i 

ft     ^        tion:    If  the  integral   \f{z)  dz  he  extended  over  a  drcuU  enclo»-    l^rU^   /r  ^ 

y  t  ing    a   point   of   discontinuity,   which   is   at   the   same   time   a 

^^■""^^-^  branch-point  at  which   m  sheets  of  the  surface  are  connected, 

^^'^"^^h^  then  the  integral  has  always  the  value  zero,  whenever  one  of  the 

''*^  ^  products  ^ 

I  C^'^^r  (^  -  V)-f{z),  iz  -  bYfiz),  ...,(.-  6)   -  fiz) 

approaches  a  finite  limiting  value. 

As  an  example,  take 

dz 


J  ^/TT^T 


<1-6U\ 


I'MTo  0, 


104  THEORY  OF  FUA^CTIONS. 

Here  f(z)  =  —  , 

V(l-;z^(l-A;V) 

which  becomes  infinite  for  2:  =  1.  This  point  is  at  the  same 
time  a  branch-point  at  which  two  sheets  are  connected.  If 
we  put  y2  =  z  —  l 

we  get      f(z)  =  <^  (0  = :  ; 

so  that  ^  =  0  is  in  fact  not  a  branch-point  for  <f>  (^). 
Now  we  get 

lim[(2-l)V(^)].=.i  =  lim 


_W(z  +  1)  (1  -  fcV). 
1 


therefore  lim  {_(z  —  l)/(i2)]a^i  =  0, 

and  hence  also 

dz 


f: 


=  0, 


V(l  -  z')  (1  -  ¥z') 

when  the  integral  is  taken  along  a  circuit  enclosing  the  point 
z  =  1.     The  integral  also  acquires  the  same  value  when  the 

circuit  encloses  one  of  the  other  branch-points  —  1,  +  -, . 

k        k 

The  investigation  of  the  value  of  the  integral  B,  in  case  the 
conditions  of  the  above  proposition  are  not  fulfilled,  must  be 
postponed  to  a  later  section  (Section  VIII.). 


SECTION  V. 

THE    LOGARITHMIC    AND   EXPONENTIAL   FUNCTIONS. 

22.  As  we  shall  be  obliged  to  make  use  of  some  of  the  prop- 
erties of  the  logarithmic  function  in  the  following  pages,  we 
must  interrupt  for  a  short  time  the  general  considerations  and 


LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS.      105 

take  up  first  the  study  of  this  special  function.  In  this  connec- 
tion, it  seems  to  us  not  unprofitable  to  make  the  investigation 
somewhat  more  exhaustive  than  would,  be  necessary  for  the 
intended  application,  and  also  to  add  directly  to  it  the  consider- 
ation of  the  exponential  function,  which  follows  from  the  log- 
arithmic. Since  we  shall  thus  have  to  deal  here  with  a  special 
case  of  the  general  investigations  to  be  taken  up  in  Sections  IX. 
and  X.,  this  example  may  also  serve  to  fix  the  ideas  for  those 
later  investigations. 

We  designate,  after  Riemann,  by  the  name  logarithm  a  func- 
tion f{z),  which  has  the  property  that 

f(zu)=f(z)+f(u).  (1) 

By  this  equation  the  function  is  entirely  determined,  except  as 
to  a  constant,  for  we  shall  be  able  to  derive  therefrom  all  its 
properties.  If,  in  the  first  place,  we  let  u  =  l,  and  leave  z 
arbitrary,  it  follows  that 

f(z)=f(z)+f{l); 
therefore  /(I)  =  Log  1  =  0. 

Again,  if  0  be  substituted  for  u,  we  have 

/(0)=/(z)+/(0); 

and  if  we  now  give  z  any  value  for  which  /(z)  is  not  zero,  it 
will  follow  that/(0)  =  Log  0  =  oo ;  for  a  similar  reason,  Log  qo 
also  becomes  infinite.  It  is  further  possible  to  express  the 
logarithm  by  an  integral ;  for,  if  equation  (1)  be  differentiated 
partially  as  to  u,  then 

zf(zu)=f'(u), 
and,  when  u  =  l, 

zfiz)  =  f'(l). 

Let  us  denote  the  constant  /'(I)  by  m.  Upon  this  constant 
depends  the  value  of  the  logarithm  of  a  number.  The  loga- 
rithms of  all  numbers  which  can  be  obtained  by  assigning  to 
the  constant   m   a   definite  value  form  together  a  system  of 


106  THEORY  OF  FUNCTIONS. 

logarithms,  and  the  constant  is  called  the  modulus  of  the  sys- 
tem of  logarithms. 
From  the  equation 

follows  d/(2)  =  d  Log  z  =  m  - ;  (2) 

z 

hence  f(z)=mC—+a 

But  since  /(I)  =  0,  the  constant  C  will  become  0,  if  1  be  taken 
for  the  lower  limit  of  the  integral,  and  z  be  made  to  assume 
real  values.     We  write,  therefore,  in  general 


T  f'dz 

Ji    z 


and  we  have  thereby  expressed  the  logarithm  by  a  definite 
integral.  For  the  purposes  of  analysis,  the  logarithms  of  that 
system  are  the  simplest  in  which  the  constant  m  assumes  the 
value  1.  These  are  called  natural  logarithms,  and  will,  in 
what  follows,  be  designated  by  the  term  log  z.     Therefore 

iogz=  r  — , 

Ji    z 
and  hence  Log  z  =  m  log  z. 

If  we  let  z  =  r  (cos  ^  -f  i  sin  <^), 

we  get 

dz  =  (cos  <f>  +  i  sin  <j>)dr  +  r  (—  sin  4>  -\-i  cos  <f>)  d(f) 
=  (cos  <f>-\-i  sin  </>)  (dr  +  ird<f>)  ; 

hence  ^  =  ^^id<fy. 

z        r 

If  z  pass  along  any  path  from  1  to  an  arbitrary  point  z,  then  r 
will  assume  the  real  values  from  1  to  r,  and  <^  those  from  0 
to  <^ ;  therefore 

J'^'dz       C^dr  ,    ., 
12      Ji    r 

or  log  z  =  log  r  +  i<j>.  (3) 


LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS.      107 

By  this  log  z  is  brought  to  the  form  of  a  complex  variable ;  for, 
since  r  assumes  only  real  and  positive  values  in  the  integral 

— ,  therefore  log  r  is  also  real ;  and  it  is  evident  that  log  r 

I     r 

is  positive  or  negative,  according  as  r  is  greater  or  less  than  1 ; 
for,  since  r  is  always  positive,  the  representing  point  moves 
along  the  positive  principal  axis,  in  the  first  case  in  the  positive 
direction,  in  the  second  case  in  the  negative ;  and  therefore  in 

the  first  case  all  elements  —  are  positive,  in  the  latter  all  are 

r 

negative. 

We  see,  further,  that  the  logarithm  depends  upon  the  path 
of  integration;  for,  let  <^  denote  the  value  acquired  by  the 
angle,  when  z  moves  from  1  to  z  along  a  line  which  does  not 
enclose  the  origin,  and  for  which  the  angles  increase,  then 
^  —  2  TT  will  be  the  value  acquired  by  this  angle  when  the  line 
moves  on  the  other  side  of  the  origin,  i.e.,  in  the  direction  of 
decreasing  angles,  from  1  to  z;  and  if  a  line  wind  n  times 
roimd  the  origin  in  the  direction  of  increasing  angles,  then  <^ 
acquires  at  z  the  value  <f>  +2  mr.    Accordingly 

log  z  =  log  r  +  i<j>  ±  2  mri. 

Our  general  considerations  are  thus  confirmed.     The  function 

-  has  no  branch-points,  but  has  the  point  of  discontinuity  z  =  0. 

z 

If  z  be  made  to  describe  a  circuit  round  the  origin,  the  value  of 
the  integral  extended  over  this  line  in  the  direction  of  increas- 
ing angles  is  2  ni,  since 

p  =  limr;z-^1      =1.  (§20) 

By  means  of  the  considerations  established  at  the  close  of 
§  20,  the  same  result  is  obtained  as  above. 

Now  from  this  it  follows  that  the  function  log  z  has  at  no 
point  of  the  plane  a  fully  determinate  value,  and  that  at  any  two 
infinitely  near  points  it  can,  by  means  of  a  suitable  arrange- 
ment of  the  path  of  integration,  acquire  values  which  differ 


108  THEORY  OF  FUNCTIONS. 

from  one  another  by  a  multiple  of  2  iri.     In  order  to  limit  as 
far  as  possible  this  indefiniteness,  we  suppose  a  line  oq  (Fig.  32), 
'q      which  does  not  cut  itself,  drawn  from 
the   origin    and   extending  to  infinity. 
Such  a  line  is  called  after  Riemann  a 
cross-cut.     Then,  of  any  two  paths  lead- 
ing from  1  to  2  and  enclosing  the  origin, 
one    must    necessarily    intersect    the 
Fig-  32.  cross-cut,    and    consequently,    on    all 

paths  not  crossing  the  cross-cut,  log  z  acquires  at  each  point 
z  a  perfectly  determinate  value,  Avhich  also  changes  every- 
where continuously  with  z.  But  at  the  points  on  the  cross- 
cut itself  the  indefiniteness  remains.  Now,  if  the  infinite 
plane  in  which  z  moves  be  designated  by  T,  and  be  supposed 
to  be  actually  cut  along  the  cross-cut  oq,  then  a  surface  arises 
which  may  be  called  T'.  In  the  latter  the  cross-cut  cannot  be 
crossed,  and  therefore  log  z  is  everywhere  a  uniform  function 
of  z  in  T',  becoming  infinite  only  for  2  =  0  and  2  =  00,  but 
elsewhere  remaining  continuous.  In  the  surface  T,  however, 
log  z  becomes  discontinuous  on  crossing  the  cross-cut.  For,  let 
Zi  and  Z2  be  two  points  on  the  two  sides  of  the  cross-cut  and 
infinitely  near  each  other  (say  Zj  on  the  right,  and  Z2  on  the 
left  of  the  direction  oq),  and  let  z  be  made  to  describe  a  closed 
line  lziZ,,cl  round  the  origin  in  the  uncut  surface  T,  starting 
from  1  and  passing  through  Zi  and  z^ ;  then,  according  to  the 
above  proposition,  the  integral 

J(lziZ2cr)  =  2Tri, 

extended  along  this  line.  But  we  have  at  the  same  time,  since 
Zi  and  Z2  ^-re  infinitely  near  each  other, 

J{lZjZ2Cl)  =  J(lZi)  -\-  J(Z2Cl)  =  J(lZi)  -  JO-CZi), 

and  consequently    J(lzi)  —  J(lcz^  =  2  -n-i. 

If,  then,  Wi  and  Wj  denote  the  values  which  log  z,  now  regarded 
as  in  T',  acquires  at  Zi  and  2:25  so  that 

Wi  =  J(lZi),    W2  =  J(1CZ2), 

we  have  Wi  —  W2  =  2  iri. 


LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS.      109 

If  the  surface  T  be  now  supposed  to  be  restored,  then  logz, 
when  z  moves  from  Zi  to  Z2,  abruptly  changes  from  Wj  into 
Wi  —  2  iri,  or  when  z  moves  from  z.^  to  Zj,  abruptly  changes 
from  w^  into  iv^  +  2  izi.  This  holds  at  whatever  place  the  path 
of  integration  may  cross  the  cross-cut.  Along  the  entire  cross- 
cut, therefore,  log  z  is  discontinuous,  the  values  of  log  z  being 
greater  by  2  7^^  for  all  points  on  the  right  side  than  for  those 
on  the  left.  This  constant  value,  by  which  all  values  of  the 
function  on  the  one  side  exceed  the  neighboring  ones  on  the 
other  side,  has  been  called  by  Riemann  the  modulus  of  peri- 
odicity of  the  function,  or  of  the  integral,  if  the  former  be  rep- 
resented by  an  integral. 

23.  The  exponential  function  can  be  derived  from  the  log- 
arithmic in  the  following  way.  By  the  symbol  a"  is  to  be 
understood  such  a  function  of  w  that 

log  (a")  =  w  •  log  a. 

Now,  if  e  denote  the  real  number  for  which  loge  has  the 
value  1,  and  accordingly  if  e  be  defined  by  the  equation 


s 


dr  _-. 
—  ^i 

1    r 


then  it  follows  that         log  (e")  =  w. 

Therefore  e"  is  the  inverse  function  of  the  logarithm;    for, 
from  e"  =  z,  follows  w  =  logz.     From  equation  (2)  (for  m  =  1) 

d  log  z  _dw_l 
dz         dz      z 

we  get  3-  =  ^  5 

dw 

consequently  —  =  t  "• 

If  we  assume  for  z  a  complex  quantity  having  the  modulus 
1,  i.e.,  if  we  let  z=  cos  <f>-\-i  sin  <j}, 


110 


THEORY  OF  FUNCTIONS. 


we  have,  in  equation  (3),  to  substitute  r  =  1,  and  therefore 
log  r  =  0.     Accordingly, 

log  (cos  <^  +  i  sin  <^)  =  i0, 

and  consequently       cos  <^  +  i  sin  <^  =  e'*. 

The  exponential  function  is  periodic ;  for,  since  to  a  value 
of  z  belongs  not  only  the  value  w,  but  also  the  values  w  ±2  mvi, 


therefore 


nio  ««i±2njr< 


and  accordingly  e"  is  not 
O   changed  if  ic  be  increased 
or  diminished  by  a  multi- 
ple of  the  modulus  of  peri- 
odicity 2  TTi.     Let  us  now 
try  to  represent  the  z^sur- 
face  T'  on  the  w;-plane  W. 
For   this  purpose  we  take  as  the  cross-cut,  for  greater  sim- 
plicity, a  straight  line  passing  through  o  and  1  (Fig.  33).     If 

z  =  r  (cos  ^  +  ^  sin  ^), 

then  w  =  log  r  -f  i^. 

Consequently  log  r  and  <^  are  the  rectangular  co-ordinates  of 
a  point  w.     Then,  if  z  be  made  to  describe  a  circle  with  radius 

1  round  the  origin  in  the 


E 


Sni 


d 


■D 


-B 


direction  of  increasing  an- 
gles from  a  to  b,  log  r  =  0, 
and  therefore  w  is  a  pure 
imaginary  and  moves  along 
the  y-axis  from  o  to  2  tti 
(Fig.  34) .  Again,  if  z  move 
from  a  along  the  left  side 
of  the  cross-cut  to  infinity, 
Pj^^  g4  <^  remains  =  0,  log  r  passes 

from  0  through  the  posi- 
tive values  to  infinity,  and  therefore  iv  describes  the  positive 
part  of  the  principal  axis.     But  if  z  move  from  a  along  the 


LOGARITHMIC  AND  EXPONENTIAL  FUNCTIONS.      Ill 

left  side  of  the  cross-cut  to  o,  then  w  describes  the  negative 
part  of  the  principal  axis  to  infinity.  But  if  z  first  arrive  at  b 
round  the  origin  on  the  right  side  of  the  cross-cut  and  then 
pass  along  its  right  side  to  oo  or  o,  iv  first  moves  on  the 
y-axis  from  o  to  2  7ri  and  then,  ^  constantly  remaining  equal 
to  2  IT,  describes  a  line  parallel  to  the  principal  axis,  first  in  the 
positive  and  then  in  the  negative  direction.  To  the  two  sides 
of  the  cross-cut  in  T',  therefore,  correspond  in  W  two  different 
lines,  i.e.,  to  the  left  side  the  principal  axis  AB,  to  the  right  a 
straight  line  CD  running  through  2  iri  parallel  to  the  principal 
axis  (Fig.  34).  If  z  be  now  made  to  pass  at  any  place  from 
the  left  side  c  of  the  cross-cut  to  the  right  side  d  by  describing 
a  circle  round  the  origin,  then  r,  and  therefore  also  logr, 
remains  constant  and  <^  increases  from  0  to  2  tt.  Consequently 
w  describes  a  line  c'd'  parallel  to  the  y-axis,  beginning  at  the 
principal  axis  AB  and  terminating  at  the  parallel  line  CD. 
It  follows,  therefore,  that  to  all  points  z  in  the  entire  infinite 
extent  of  the  surface  T',  in  which  <^  cannot  increase  beyond 
2  IT,  corresj)ond  only  such  points  w  as  lie  within  the  strip 
formed  by  the  two  parallel  lines  AB  and  CD.  The  function 
e",  or  z,  thus  assumes  within  this  strip  all  its  possible  values, 
and,  indeed,  each  bvit  once,  since  to  any  two  different  values  of 
w  =  log  r  +  i<f>  belong  also 
different  values  of  r  and 
<^,  and  therefore  also  dif- 
ferent values  of 

e"  =  z  =  r  (cos  <^  +  ?'  sin  </>). 

If  we  wish  to  bound 
the  surface  T',  this  can  be 
effected,  on  the  one  hand, 
by   describing  round  the 

origin  a  circle  with  a  very  small  radius  p.  To  this  corresponds 
in  W,  since  p  remains  constant,  a  line  ns  running  parallel  to 
the  ?/-axis  between  the  two  parallel  lines  AB  and  CD,  and 
very  far  removed  from  the  origin  on  the  negative  side.  This 
moves  to  infinity  when  p  tends  towards  zero,  i.e.,  when  the 


E- 
C- 


-T> 


c 


112  THEORY  OF  FUNCTIONS. 

circle  shrinks  into  the  origin.  At  all  points  of  this  line  ns, 
which  has  been  removed  to  infinity,  e"'  has  therefore  the  value 
zero.  On  the  other  hand,  the  boundary  of  T'  can  be  formed 
by  a  circle  round  the  origin  with  a  very  large  radius  B. 
To  this  corresponds  in  w  a  straight  line  mr  on  the  positive 
side,  which  is  very  far  removed  and  is  parallel  to  the  ?/-axis. 
If  jB  increase  indefinitely,  this  straight  line  also  moves  to 
infinity,  and  at  all  points  on  it  e*"  is  infinite.     The  surface 

T'  can  be  assumed  closed 
^ F  at  infinity ;  then  the  cir- 
cle with  the  large  radius  M 
is  represented  by  a  small 
■B  circle  round  the  point  oo, 
which  shrinks  into  this 
point  when  R  increases 
indefinitely.  Therefore 
the  two  sides  of  the  cross- 
cut extending  from  o  to 
00  form  alone  the  boundary  of  a  spherical  surface  T',  and  to 
the  latter  corresponds  the  strip  between  the  parallel  lines  AB 
and  CD  extending  on  both  sides  to  infinity. 

If  we  now  increase  the  angle  <f)  beyond  2  tt,  the  function  w, 
or  log  z,  proceeds  continuously.  Then  the  cross-cut  can  be  sup- 
posed to  be  like  a  branch-cut,  across  which  the  surface  T'  is 
continued  into  another  sheet.  In  this  second  sheet,  then,  all 
relations  are  the  same  as  in  the  first,  except  that  at  all  points 
in  it  <^  is  greater  by  2  tt,  and  accordingly  w  by  2  iri,  than  at  the 
corresponding  places  in  the  first  sheet.  Therefore  we  obtain  a 
second  strip  between  the  parallel  lines  CD  and  EF,  which  pass 
through  2  iri  and  4  Trt.  By  continuing  this  mode  of  treatment 
and  applying  it  also  to  negative  values  of  <f>,  the  plane  W  is 
divided  into  an  infinite  number  of  parallel  strips.  In  each  of 
them  the  function  e'"  assumes  all  its  values  once  and  has  the 
same  values  at  any  two  corresponding  points  of  two  different 
strips.  On  the  positive  side  of  each  strip  e^  tends  towards 
infinity,  but  on  the  negative  side  it  approaches  zero. 


Fig.  84. 


GENERAL  PROPERTIES  OF  FUNCTIONS.  113 

sectio:n^  VI. 

GEXERAL    PROPERTIES    OF    FUNCTIONS. 

24.   The  basis  for  the  following  investigations  is  found  in 
the  exceedingly  important  proposition  proved  in  §  20 :  If  the 

integral   \f{z)dz  be  extended  over  the  boundary  of  a  region 

in  which  f{z)  becomes  discontinuous  only  at  a  point  z  =  a, 
which  is  not  a  branch-point,  and  in  such  a  manner  that 
(z  —a)f(z)  approaches,  for  z  =  a,  a  definite  finite  limiting  value 
p,  independent  of  the  mode  of  approach  to  a,  then 


if(z)dz  =  2  Trip. 


Now,  if  <f>  (z)  be  a  function  which  possesses  no  branch-points 
in  a  region  T,  and  which  remains  finite  and  continuous  both  in 
the  interior  and  along  the  boundary  of  T,  and  if  t  denote  an 
arbitrary  point  in  this  surface,  then  the  function 

has  in  T  the  properties  required  in  the  above  proposition.  It 
becomes  discontinuous  only  for  z  —  t,  and  since,  like  (^(z),  it 
possesses  no  branch-points  within  T,  t  can  never  fall  on  such  a 
point ;  further,  /  (z)  becomes  discontinuous  for  z  =  tva.  such  a 
way  that 

{z-t)f{z)  =  ^{z) 

tends  towards  a  definite  finite  limiting  value,  namely  </>  (f). 
Therefore 


and  consequently 


P- 


z—t 

4.(f)=^.c-m^,  (1) 

Z  TTltJ        Z  I 


the  integral  being  extended  over  the  boundary  of  T. 

The  validity  of  this  equation  is  conditioned  upon  the  suppo- 
sition that  the  function  ^  (t),  which  is  uniform  and  continuous 


114  THEORY  OF  FUNCTIONS. 

within  T,  has  a  fully  determinate,  finite  value  at  any  point  t 
of  this  region,  the  value  being  always  the  same,  however  the 
variable  may  approach  this  point.  It  may  therefore  be  here 
noted  that,  in  the  case  of  uniform  functions,  this  condition  is 
not  always  satisfied^  at  special  points,  but  at  such  points  the 
function  is  always  at  the  same  time  discontinuous.  For 
instance,  since  the  function  e',  for  z  =  cx^,  becomes  zero  or 
infinite,  according  as  the  variable  z  passes  to  infinity  through 
negative  or  through  positive  values  (§  23),  therefore  the  func- 
tion ei,  for  2=0,  acquires  the  value  zero  or  becomes  infinite, 
according  as  z  approaches  zero  through  the  real  negative  or 
through  the  real  positive  values.     Likewise  the  function 

1' 
c  —  e' 

in  which  c  denotes  a  constant,  assumes  for  z  =  0,  in  the  former 
case  the  value  c,  in  the  latter  case  the  value  zero.  At  such  a 
point,  however,  the  continuity  also  always  ceases.  For,  if  in 
the  above  example  the  variable  be  made  to  increase  through 
the  real  values,  the  function,  at  the  passage  through  the  value 
z==0,is  suddenly  changed  from  c  into  0. 

Thus  the  requirement  that  a  function  be  everywhere  continuous 
in  a  region,  at  the  same  time  excludes  the  occurrence  of  such 
points. 

Now,  if  the  above  conditions  be  fulfilled,  equation  (1)  gives 
the  value  of  the  function  <j>  at  any  point  t  in  the  interior  of  T 
by  an  integral,  in  which  the  variable  z  passes  through  only  the 
points  on  the  boundary  of  T;  this  integral  has  indeed  a  finite 
Ly^  value   at   every   point   t  situated  in  the   interior   of  T,   and 

D  changes  continuously  with  t,  as  will  be  proved  later.     Let  the 

function  <fi  (z)  be  given,  not  by  an  expression,  but  by  its  values 
for  the  points  of  a  certain  region;  then  it  follows  from  the 

1  This  multiplicity  of  values  of  tlie  function  has  nothing  in  common 
with  that  discussed  in  Section  III.,  which  is  brought  about,  in  the  case  of 
multiform  functions,  by  a  multiplicity  of  paths.  By  the  introduction  of 
Riemann  surfaces  this  kind  of  multiplicity  is  removed. 


GENERAL  PROPERTIES   OF  FUNCTIONS. 


115 


above  equation  that,  if  the  function  be  given  only  for  all  points 
of  the  boundary  of  T,  it  can  also  be  ascertained  for  all  points 
in  the  interior  of  T,  and  consequently  cannot  longer  be  arbi- 
trarily assumed  in  the  interior  of  T. 

For  example,  if  a  function  <f>  (z)  have  everywhere  along  the 
boundary  of  T  the  constant  value  C,  we  obtain  from  (1) 


27riJz—t      2-771 J  z~t 


But  this  integral  retains  its  value  if  the  curve  of  integration  be 
replaced  by  a  circle  described  round  t.    Then  we  have  (§  20) 

dz       o    . 


/. 


z—t 
and  consequently,  for  every  value  of  t. 

Therefore,  if  a  function  be  uniform  and  continuous  every- 
where in  a  region  T,  and  if  it  have  the  constant  value  C  along 
the  boundary  of  T,  it  is  also  constantly  equal  to  C  everywhere 
in  the  interior  of  T.  It  follows,  further,  from  (1),  by  differen- 
tiation as  to  t,  that 


<^<"^(0  = 


2.3 


2  7ri 


'■fl 


<t>(z) 


-,dz. 


(2) 


{z  -  ty+^ 

All  these  integrals  extend  over  the  boundary  of  T,  while  t 
lies  in  the  interior  of  T;  consequently  in  them  z  —  t  never 
vanishes.     Therefore,  if  h  denote  any  positive  integer,  then 

1 

{z  -  ty 


116  THEORY  OF  FUNCTIONS. 

is  finite  for  every  value  of  t  considered,  and  changes  continu- 
ously with  t.  The  same  holds  if  the  above  fraction  be  multi- 
plied by  any  value  ^  (z)  which  is  independent  of  t ;  consequently 
the  sum  represented  by  the  integral 

•<^  (z)  dz 


f\ 


(z-tf 

in  which  <f>  (z)  has  to  assume  in  succession  all  the  values  occur- 
ring along  the  boundary  of  T,  also  changes  continuously  with 
t.  And  since  these  values  are  finite  according  to  the  assump- 
tion, the  integral  has  also  a  finite  value  (§  16).  Accord- 
ingly all  the  above  integrals,  as  well  as  those  contained  in  (1), 
are  finite  and  continuous  functions  of  t  within  T^.  From 
this  follows  the  proposition:  If  a  function  have  no  brancJir 
points  in  the  interior  of  a  region  and  be  finite  and  continuous 
therein,  then  all  its  derivatives  in  the  same  region  are  also  finite 
and  continuous. 

If  in  equation  (1)  the  integration  be  referred  to  an  arbitrarily 
small  circle  round  the  point  t  with  radius  r,  and  if  for  this  pur- 
pose we  let 

z  —  t=^r  (cos  6  +  i  sin  &), 

then  =  idO, 

z  —  t 

1   r^ 

and  hence  ^(t)  =  -—  I     <^  (z)  dO. 

If  we  now  let 

^{z)  =  u-\-  iv,   <f>(t)=U(,-}-  ivo, 

we  obtain,  on  separating  the  real  from  the  imaginary, 

Uo  =  ~-  )     ud6,  -^0  =  --  I     vdO. 
2  IT  Jo  ZttJo 

Hence  it  follows  that  the  real  components  of  the  function  <^ 
are,  at  the  point  t,  the  mean  values  of  all  the  surrounding 
adjacent  values  of  these  components.     Therefore  Uq  must  be 

1  C.  Newmann,  Vorlesungen  uber  Biemanri's  Theorie  der  AbeVschen 
Integrale,  S.  91. 


GENERAL  PROPERTIES   OF  FUNCTIONS.  117 

greater  than  one  part  and  at  the  same  time  less  than  another 
part  of  these  adjacent  values.  The  same  conclusion  holds  for 
Vo]  and  since  it  also  holds  at  each  point  of  the  surface  T, 
the  real  components  of  the  function  <^  do  not  have  a  maximum 
or  a  minimum  value  at  any  point  in  T. 

25.  By  means  of  equation  (1)  the  function  <^  can  be  devel- 
oped in  a  convergent  series.  Let  us  describe  round  an 
arbitrary  point  a  of  the  region  T  a  circle,  which  is  still 
wholly  within  this  region,  and  therefore  does  not  extend  quite 
to  the  branch-point  or  point  of  discontinuity  nearest  to  a ;  and 
let  us  first  take  this  circle  as  the  curve  of  integration  in  equar 
tion  (1).     Now,  for  every  point  t  lying  within  the  circle, 

mod  (z  —  a)  >  mod  (t  —  a) 

(Fig.  35),  since  z,  during  the  integration,  passes  through  only 
points  on  the  circumference  of  the  circle ;  therefore  az  >  at. 
We  can  also  put 

1     _  1  1  1 

z  —  t     z  —  a—  (t  —  a)     z  —  a  ^  _t  —  a 

z  —  a 

and  since  mod <  1, 

z  —  a 

we  can  develop  this  fraction  in  the  convergent  series 

z  —  t     z  —  a  i        z  —  a     {z  —  ay     {z  —  ay  ' 

If  this  series  be  substituted  in  (1),  we  get 

which  is  the  same  as  Taylor's  series;  for,  according  to  (1), 

1    r*(?l*=^(a),  (4) 

2  iTiJ    z  —  a 


118  THEORY  OF  FUNCTIONS. 

and,  according  to  equations  (2), 

1     r  <f>(z)dz   _  <^^">(a) 


iJ  (z  —  d 


27rtJ  (2-a)"+i     2'Z  —  n 
consequently  we  obtain 


(6) 


(6) 


This  method  of  deriving  Taylor's  series  has  the  advantage 
of  showing  exactly  how  far  the  converge  ncy  of  the  series 

extends,  namely,  to  all  points  t 
which  are  at  a  less  distance  from 
a  than  the  nearest  point  of  discon- 
tinuity or  branch-point.  In  Fig. 
35  three  such  points  are  marked 
by  crosses.  The  above-mentioned 
circle  described  round  a,  of  which 
the  radius  is  so  chosen  that  there 
is  no  point  of  discontinuity  or 
branch-point  within  it  or  on  its 
circumference,  is  called  the  domain 
of  the  point  a.  The  following  prop- 
osition can  then  be  enunciated:  If 
a  function  <^  (i)  he  finite  and  continuous  at  a  point  a  which  is  not 
a  branch-point,  then,  for  any  point  t  in  the  domain  of  a,  it  can  be 
represented  by  a  convergent  series  of  ascending  powers  of  t  —  a; 
for,  if  we  let 

_  1    r  ct>(z)dz 

"     2TTiJ  (z-a)"+^' 


Fig.  35. 


P. 


in  which  the  integration  is  to  be  extended  either  along  the 
above  circle  or  along  any  other  line  surrounding  the  point  a 
and  enclosing  no  point  of  discontinuity  or  branch-point,  then 

by  (3) 

<f> (t)  =  Po+ Pi (t  -  a) -\-p,(t  -  ay  -\-  •••,  (7) 


GENERAL  PROPERTIES   OF  FUNCTIONS.  119 

in  whiclx,  according  to  page  116,  all  the  coefficients  p  have 
finite  values. 

Though  in  the  series  (3)  all  integrations  must  at  first  be 
taken  along  the  circle  described  round  a,  yet,  since  the 
functions 

<^(^)       <f>(^)         ^(^)      etc 
z  —  a   (z  —  ay   (z  —  ay 

remain  finite  and  continuous  up  to  the  point  a,  the  integrals 
can  also  be  taken  along  any  arbitrarily  small  circle  described 
round  a,  without  changing  their  values.  It  follows  that,  if  the 
function  <^  be  given  by  its  values  in  an  arbitrarily  small  finite 
region  containing  the  point  a,  then  all  those  integrals,  and, 
consequently,  all  coefficients  of  the  convergent  series,  are 
thereby  determined,  and  therefore  the  value  of  the  function 
for  any  point  within  the  large  circle  can  be  ascertained. 

Now  let  tti  be  a  point  which  still  lies  within  this  circle,  then 
^(t)  will  be  known  both  at  aj  and  also  in  the  region  immedi- 
ately contiguous  to  ttj.  Then  if  a  circle  be  described  round  ai, 
which  still  leaves  outside  all  points  of  discontinuity  and  branch- 
points (Fig.  35),  <fi(t)  can  be  developed  in  a  new  series  for 
all  points  in  this  circular  region.  It  is  evident  that  by  contin- 
uing in  this  way  the  function  4*(t),  which  is  given  only  within 
an  arbitrarily  small  finite  part  of  a  region  T,  can  then  be  deter- 
mined in  the  whole  region  T,  when  this  contains  neither  a 
point  of  discontinuity  nor  a  branch-point. 

The  same  holds  if  the  function  </>(<)  be  given  only  along  an 
arbitrarily  small  finite  line  proceeding  from  a.  For,  if  this  be 
the  case,  let  us  denote  the  continuously  successive  points  of 
this  line  by  a,  h,  c,  d,  etc. ;  then 

^^  ^  b-a 

is  therefore  known,  if  <^(a)  and  <f>(b)  be  known.     Likewise 

</.'(fi)  =  lim  '^(^)-<^(^), 


120  THEORY  OF  FUNCTIONS. 

by  which  ^'(&)  is  determined.  In  this  manner  the  values  of 
the  derivatives  ^'(f),  for  all  points  a,  h,  c,  d,  etc.,  can  be  found. 
Then 

^"(a)=lim^M^M^, 
b  —  a 

^ ^ '  c-h        ' 

etc., 

so  that  the  second  derivatives  are  also  known.  By  continuing 
in  this  way  we  can  determine  the  values  of  all  derivatives  for 
the  point  a,  and  consequently  of  all  the  coefficients  of  the 
series  (6).  We  then  obtain,  for  every  point  t  within  the  first 
circle,  an  expression  for  <^{t)  in  the  form  of  a  convergent  series. 
Accordingly  we  can  continue  as  above  and,  starting  from  a 
small  region  containing  the  point  Oi,  ascertain  the  value  of  ^(?) 
for  all  points  in  the  second  circle,  etc.  From  the  above  fol- 
lows the  proposition :  A  function  of  a  complex  variable,  ichich  is 
given  in  an  arbitrarily  small  finite  portion  of  the  z-plane,  can  be 
continued  beyond  it  in  only  one  way.  As  a  special  case  of  this 
proposition  we  emphasize  the  following :  If  a  function  be  con- 
stant in  a  finite  arbitrarily  small  portion  of  the  region  T,  then  it  is 
constant  everywhere  in  T.  For,  if  it  always  equal  C  in  a  small 
portion  of  the  surface  containing  the  point  a,  let  us  take  a 
circle,  described  round  a  and  lying  within  this  small  region, 
as  the  curve  of  integration  in  equations  (4)  and  (5),  and  let 

z  —  a=r  (cos  6  +i sin  6)  ; 

then  it  follows  from  (4)  that 

<^(«)  =  ^)    <l>(^)de  =  ^i   dd=c, 

since  ^(z)  possesses  the  value  C  at  all  points  on  the  circum- 
ference of  the  circle.     Further,  (5)  becomes 


^^^J_  p    c^(^)     aQ^l_q_  p(cosne-*sinn^)d^, 


GENERAL  PROPERTIES  OF  FUNCTIONS.  121 

and  this  value  vanishes,  since,  for  every  integral  value  of  n 
different  from  zero, 

J  I  '"  cos  n6de  =  0  and   f"  sin  nO  d6  =  0. 
0  Jo 

Hence,  in  the  series  (3),  ^(a)  becomes  equal  to  C,  and  all 
other  terms  disappear ;  consequently,  for  any  point  of  the  circle 
of  convergence,  ^(?)  is  equal  to  C.  If  the  function  be  con- 
tinued in  the  manner  indicated  above,  </>(^)  remains  every- 
where constantly  equal  to  C.  The  same  holds  if  <^{t)  be 
constant  along  an  arbitrarily  small  finite  line.  In  this  case, 
the  above  notation  being  employed,  the  values  <^(a),  ^(6),  <^(c), 
etc.,  are  all  equal  to  C,  and  thus  all  the  derivatives  «/>'(«)>  ^"(«)> 
etc.,  again  vanish,  and  thereby  also  all  coefficients  of  the  series 
(6)  except  the  first,  which  is  equal  to  C.  The  same  holds 
therefore  as  above. 

From  this  special  proposition  can  again  be  deduced  the 
preceding  more  general  one.  For,  if  two  functions  ^(t)  and 
ifiit)  agree  in  their  values  in  an  arbitrarily  small  portion 
of  a  region  or  of  a  line,  then  in  this  portion  the  fimction 
tj>(t)  —  (/'(<)  is  constantly  equal  to  zero ;  consequently  this 
function  is  everywhere  equal  to  zero,  i.e.,  \p(t)  is  everywhere 
equal  to  ^(t),  and  therefore  the  function  </)(f)  cannot  be  con- 
tinued in  two  different  ways  from  the  portion  in  which  it 
is  given. 

26.  We  now  proceed  to  represent  a  function,  which  suffers 
a  discontinuity  of  any  kind  whatever  at  a  point  a  (not  a  branch- 
point), by  a  series  in  the  domain  of  this  point. 

Let  two  circles  be  described  round  the  point  a  as  centre; 
call  the  smaller  C,  the  larger  K.  We  assume  that  the  function 
<f>{t)  does  not  possess  a  branch-point,  either  within  the  smaller 
circle  or  in  the  ring  formed  by  the  two  circles;  further,  let 
<f>{t)  be  continuous  everywhere  within  the  ring,  but  on  the 
other  hand  possibly  become  discontinuous  in  any  way  what- 
ever within  C.  Then  the  two  circles,  C  and  K,  bound  a  region 
in  which  <f>  (t)  satisfies  all  the  conditions  under  which  equation 


122  THEOBY  OF  FUNCTIONS. 

(1),  §  24,  holds.  We  have  therefore,  at  every  point  t  in  the 
interior  of  the  ring, 

^^^        2  7riJ      Z-t 

wherein,  however,  the  integral  must  be  extended  round  each 
of  the  circles  in  the  positive  boundary-direction,  and  hence 
round  the  small  circle  in  the  direction  of  decreasing  angles. 
Therefore  we  can  put 

^^^     2TriJ    z-t       2TriJ    z-t 

Then  the  first  integral  refers  to  the  circle  K,  the  second  to  ,C, 
and  both  are  to  be  taken  in  the  direction  of  increasing  angles. 
Since,  for  every  point  t  in  the  interior  of  the  ring, 

mod  (t  —  a)  <  mod  (z  —  a), 

the  first  integral  Ji  furnishes  the  same  development  as  was 
derived  in  §  25.     We  thus  obtain  by  (7) 

Ji  =Po  +Pi («  -  a)  +P2 (t  -  af  +P3{t-  af  -\ , 

wherein  «_  = |  -^U .  (8) 

For  the  second  integral  J2,  on  the  other  hand,  all  the  points 
t  within  the  ring  lie  outside  the  circle  C  described  by  the 
variable  z ;  hence  in  this  case  az  <  at,  or 

mod  (2  —  a)  <  mod  (t  —  a)  and  mod <  1. 

t  —  a 

Therefore,  if  we  put 

1  1  11 


—  t     t  —  a—  (z  —  a)      t  —  a  -i 


t  —  a 

we  can  develop  this  fraction  in  a  convergent  series  of  ascend- 
ing powers  of     ~    ,  and  we  obtain 

t  —  0/ 

1     _     1  z  —  a       (z  —  ay 

z^t~  t  -  a      {t  -  af      (t  -  af 


GENERAL  PROPERTIES  OF  FUNCTIONS.  123 

If  we  substitute  this  value  in  J.,,  we  get 

or  if,  for  the  sake  of  brevity,  we  let 

(9)  ^C<f>(z)(z-ay-'dz  =  c^''\ 

'     t-a      (t~ay^(t-ay^ 
Hence  we  obtain  for  all  points  t  within  the  ring  the  series 

(10)  <f> (t)  =p,  +  p,(t-  a)  +p,(t-  ay  +p^(t-ay+-- 

+  _±_-l 2 !___£ 4..... 

^t-a     (t-ay     {t-ay 

This  development  can  be  applied  when  a  function  <f)  (t)  suf- 
fers a  discontinuity  of  any  kind  whatever  at  a  point  a,  which 
is  not  a  branch-point.  For,  enclosing  the  point  of  discon- 
tinuity a  in  an  arbitrarily  small  circle,  the  hypotheses  pre- 
viously made  are  conformed  to,  if  we  take  this  circle  as  the 
curve  of  integration  C  for  the  integrals  c^"\  and  refer  the  inte- 
grals p„  to  a  circle  K,  which  is  only  so  large  that  every  other 
point  of  discontinuity  occurring  (besides  a),  and  every  branch- 
point, lies  outside  K.  Then  series  (10)  furnishes  a  finite  value 
for  (f>  (t)  at  every  point  t  lying  within  K,  with  the  exception  of 
the  point  a  itself.  We  remark  in  this  connection  that  the 
integrals  c<"^  can  also  be  taken  along  the  circle  K,  since  they 
have  the  same  values  for  it  as  for  the  circle  C  (§  19). 

From  the  preceding  can  be  derived  also  a  series  which  holds 
when  «^(^)  suffers  any  discontinuity  at  the  point  f  =  co,  and 
when  that  point  is  not  a  branch-point.     To  this  end  we  let 

Z  =  -5     t  =-y 
U  1' 


124  THEORY  OF  FUNCTIONS. 

whereby  ^  (z)  changes  into  f(u),  say,  and  <^  (t)  into  f(v)  ;  then 
f(v)  is  discontinuous  for  v  =  0.  Now  let  z  describe  a  circle  K 
round  the  origin,  and  accordingly  let 

z  =  r  (cos  6  +  i  sin  6), 

then  u  =  -  (cos  6  —  i  sin  ^) ; 

r 

hence  u  likewise  describes  a  circle  (/round  the  origin,  but  in  the 

opposite  direction.     Since,  further,  -  decreases  as  r  increases, 

r 

to  the  points  z  lying  outside  Z  correspond  the  points  u  lying 
within  U.  Therefore,  if  we  assume  the  circle  Z  so  large  that 
it  encloses  all  branch-points,  and  that  <f>(t)  is  discontinuous 
outside  Z  only  for  t  =  cc,  then  f(v)  has  no  branch-point  within 
U  and  suffers  a  discontinuity  only  for  v  =  0.  Hence  ^  we  can 
use  series  (10)  for  the  development  of  f(v),  if  we  put  a  =  0, 
and  we  obtain 

c'      c"      c'" 
(11)    f(v)=po-\-piV+i)2V^+p^v^-\ 1 1--^  +  ^  +  •••, 

wherein,  by  (8)  and  (9), 

Both  integrals,  according  to  the  remark  made  above,  can  be 
taken  round  the  circle  U,  which  in  this  case  takes  the  place  of 
the  circle  K;  they  are  to  be  taken,  like  (8)  and  (9),  in  the 
direction  of  increasing  angles.  If  we  introduce  z  and  t  again 
in  place  of  u  and  v,  then 

,  dz 

du  =  -  —  ; 

z^ 

therefore  J±  =  -z-~^dz,   u^-Hu  =  -—- 

1  We  remark  that,  since  m  =  0  is  not  a  branch-point  according  to  the 
assumption,  we  can  so  draw  the  branch-cuts  that  none  of  them  meet 
the  point  m  =  0  ;  then  the  line  CT,  and  therefore  also  the  line  Z,  bounds  a 
portion  of  the  surface. 


INF.  AND  INF'L    VALUES   OF  FUNCTIONS.        125 

The  integrals,  to  be  taken  as  to  z,  are  then  extended  round  the 
circle  Z,  but  in  the  direction  of  decreasing  angles,  since  U  was 
described  in  the  opposite  direction.  If  we  wish  to  take  them 
also  in  the  direction  of  increasing  angles,  we  have  to  erase  the 
minus  signs,  and  we  then  obtain 

and  hence  from  (11) 

(13)      ^(t)  =p,+^+^^-\.^  +  ...  +  c't  +  c"f  +c"'f  +  .-. 

This  series  represents  the  value  of  <^  (t)  at  all  points  t  (except 
t  =  <X))  which  lie  outside  such  a  circle  Z,  described  round  the 
origin,  that  all  finite  points  of  discontinuity  and  all  branch- 
points are  situated  within  the  same. 


SECTION  VII. 

INFINITE    AND    INFINITESIMAL   VALUES    OF    FUNCTIONS. 

A.   Functions  without  branchpoints.     Uniform  functions. 

27.  In  the  closer  examination  of  points  of  discontinuity,  to 
which  we  now  turn,  we  shall  at  first  entirely  exclude  branch- 
points from  our  considerations.  These  therefore,  in  general, 
relate  to  uniform  functions,  yet  it  may  be  expressly  stated 
that  they  hold  also  for  multiform  functions,  as  long  as  the 
discussion  refers  to  only  finite  parts  of  the  plane  in  which 
there  are  no  branch-points. 

If  we  let  the  variable  z  approach  a  point  a,  a  function  ^  (z) 
either  does  or  does  not  receive  the  same  value  for  all  paths  of 
approach ;  and,  in  the  former  case,  the  acquired  value  can  be 
either  finite  or  infinite.  Hence  there  are,  for  the  behavior  of 
a  function  ^  (2;)  at  a  point  a,  the  following  possibilities,  and 
only  these  -.  — 


126  THEORY  OF  FUNCTIONS. 

(1)  The  function  axjquires  at  a  for  all  paths  of  approach  to 
this  point  one  and  the  same  finite  value. 

(2)  The  function  becomes  infinite  at  a  for  all  paths  of 
approach. 

(3)  The  function  does  not  acquire  at  a  the  same  value  for 
all  paths  of  approach,  but  can  for  different  paths  receive 
different  values.^  (That  this  can,  in  fact,  occur  has  been  shown 
already  by  examples  [§  24].) 

In  the  first  case,  and  only  in  this,  is  the  function  continuous  at 
the  point  a;  in  the  two  other  cases  it  is  discontinuous.  There 
are  therefore  two,  and  only  two,  different  kinds  of  discon- 
tinuity, and  these  are  also  distinguished  by  special  names. 

By  a  discontinuity  of  the  first  kind,  or  a  polar  discontinuity,^ 
we  understand  the  case  when  a  function  <^  (z)  becomes  infinite 
at  a  for  every  path  of  approach  of  the  variable  to  this  point. 
Such  a  discontinuity  is  characterized  also  by  the  condition 

that  — — -  is  absolutely  continuous  at  z  =  a,  and  that  therefore 

it  acquires  the  value  zero  for  every  path  of  approach  to  the 
point  a. 

A  discontinuity  of  the  second  kind,  or  a  non-polar  discontinuity, 
occurs  at  a  point  a  when,  on  the  contrary,  the  value  acquired 
by  the  function  at  a  can  be  different,  according  to  the  path 
and  manner  of  approach  of  the  variable  to  the  point  a.  For 
instance,  if  a  line  map  can  be  drawn  through  a  so  that  the 
function  acquires  for  the  path  ma  a  value  different  from  that 
for  the  path  pa,  then  the  function  springs  abruptly  from  the 
former  value  to  the  latter,  when  z  passes  through  a  on  the  line 

1  We  might  also  think  it  possible  that  the  function  could  become 
infinite  of  different  orders  at  a  for  different  paths  of  approach.  But«  in 
addition  to  the  fact  that  this  will  later  be  proved  to  be  impossible,  such 
a  case  cannot  be  taken  into  consideration  at  present,  because  the  con- 
ception of  infinity  of  any  definite  order  cannot  yet  be  introduced.  The 
question  at  present  is  rather  only  the  alternative,  whether,  if  the  function 
acquire  at  a  the  same  value  for  all  paths  of  approach,  this  value  is  finite 
(zero  included),  or  not. 

2  C.  Neumann,  Vorlesungen  uber  Biemanii's  Theorie  der  Abel'schen 
Funktionen,  S.  94. 


INF.  AND  INF'L    VALUES  OF  FUNCTIONS.        127 

map,  and  thereby  suffers  a  discontinuity  of  the  second  kind. 
Such  a  discontinuity  occurs  in  e*  for  2  =  cc,  since  e*  becomes 
infinite,  zero,  or  indeterminate,  according  to  the  direction  in 
which  z  moves  away  to  infinity.  For,  let  z  =  r  (cos  (f>  +  i  sin  <j>), 
then  only  r  becomes  infinite,  while  ^  indicates  the  direction  in 
which  z  moves  away  to  infinity.     Then  we  obtain 

gz  ^  grco,0  g.vsin,i,  ^  e"°'*[cos(r  sin  <^)  +  i  sin  (r  sin  <^)], 

wherein  the  second  factor  always  maintains   a  finite   value. 

When  r  becomes  infinite,  however,  the  first   factor  becomes 

infinite  or  zero,  according  as  cos  «^  is  positive  or  negative.     If, 

on  the  other  hand,  cos  ^  =  0,  then  r  cos  <f>,  and  therefore  also 

1 

the  first  factor,  is  quite  undetermined.  In  the  function  e* 
occurs  likewise  a  discontinuity  of  the  second  kind  for  z  =  0. 

An  important  property,  manifesting  itself  at  places  of  dis- 
continuity of  the  second  kind,  results  from  the  following  con- 
siderations. If  a  function  ^(2)  be  absolutely  continuous,  and 
hence  also  not  infinite  at  a  point  a,  the  product  (z  —  a)<i>(z) 
acquires  the  value  zero  at  a  for  all  paths  of  approach.  We 
will  now  show  that  the  converse  also  holds,  namely,  that  if 

lim  (z  —  a)^(z)=  0, 

for  all  paths  of  approach  to  the  point  a  (which,  as  is  always 
assumed  here,  is  not  a  branch-point),  the  function  <l>{z)  must 
be  continuous  at  a.  For  {z  —  a)(t>(z)  is,  according  to  the 
assumption,  continuous  at  a,  and  hence  can  be  represented  by 
a  series  of  ascending  powers  oiz  —  a  converging  for  all  points 
z  in  the  domain  of  a  (§  25).     Let 

(z  -  a)ti>{z)  =  Po-\-Pi(z  -  a)  +p4.z  -  a^-^-Pziz  -  a)^  + . . .. 

Therein  i\  denotes  the  value  of  {z  —  a)^{z)  for  z  =  a ;  and  since 
this  is  zero  according  to  the  hypothesis,  it  follows  that 

(z  -  a)«^(z)  =i)i(z  -  a)  +i>2(2;  -  of  ^Pz{z  -  af  +  •••, 

from  which  is  obtained 

<^(2)  =Pi  +P2(z  -  a)  +ps(z  -ay  +  '-. 


128  THEORY  OF  FUNCTIONS. 

Accordingly  <l>{z)  assumes  the  finite  value  pi  for  all  paths  of 
approach  to  the  point  a,  and  it  is  therefore  continuous  at  a. 
We  thus  obtain  the  following  proposition :  The  necessary  and 
sufficient  condition  to  ensure  that  a  uniform  function  <f>{z)  is 
finite  and  continuous  at  a  point  a  is 

Urn  [(z  —  a)<^(z)],^„  =  0. 

If  we  put  (2  —  a)<f>(z)  =  F(z),  we  can  express  this  proposition 
also  in  the  form :    If  the  function  F(z)  have  the  value  zero  at 

a  for  all  paths  of  approach  to  this  point,  then  ^  '  is  continuous 
at  a ;  and  conversely. 

From  this  now  follows :  If  a  function  <f>(z)  suffer  a  non- 
polar  discontinuity  at  a  point  z  =  a,  it  must  dl,so  become  infinite 
for  some  manner  of  approach  to  a.  For,  if  ^(z)  were  to  acquire 
at  a  for  different  paths  of  approach  values  not  only  different 
but  finite,  then  would 

lim  [(z  -  a)<^(z)],i<.  =  0 

for  all  paths  of  approach,  and  ^(z)  would  not  suffer  any  dis- 
continuity at  a.  Since  now  the  function  always  becomes 
infinite  for  a  discontinuity  of  the  first  kind,  we  can  express 
the  preceding  proposition  also  in  this  way  :  A  uniform  function 
can  he  discontinuous,  only  when  at  the  same  time  it  becomes 
infinite;  for,  in  the  case  of  a  polar  discontinuity  this  always 
occurs,  and  for  a  non-polar,  at  least  by  one  way  of  approach. 

But  the  fimction  must  be  capable  of  assuming  any  arbitrarily 
assigned  value  at  a  point  of  discontinuity  of  the  second  kind  a. 
For,  if  c  be  such  a  value,  and  if  <^(z)  suffer  a  non-polar  discon- 
tinuity at  a,  so  do  also  «i(z)  —  c  and  ,  because  these 

"^  '  ^^^  <^(z)-c 

functions  likewise  acquire  different  values  for  different  paths 
of  approach  to  a,  when  this  is  true  of  ^(z).  !Now  since  these 
functions  must  also  once  become  infinite,  ^(z)  —  c  must  once 
become  zero,  and  therefore  <^(z)  must  be  equal  to  c  for  some 
one  way  of  approach. 

We  will  make  this  clear  by  an  example,  and  in  this  special 
case  seek  to  determine  also  what  must  be  the  way  of  approach 


INF.  AND  INF'L    VALUES   OF  FUNCTIONS.         129 

to  ensure  that  a  function  acquires  an  assigned  value.  To  this 
end  we  shall  consider  the  function  already  instanced  (p.  114), 

1' 
c  —  e' 

in  which  c  denotes  an  arbitrary  constant.     This  function  has  a 

discontinuity  of  the  second  kind  at  the  point  2  =  0.     Since  it 

1 

must  also  become  infinite  here,  e'  must  be  capable  of  assuming 

the  arbitrary  value  c  for  2  =  0.     We  will  inquire  when  this 

takes  place.     Kot  to  disguise  the  general  nature  of  the  process 

by  special  circumstances,  we  will  assume  c  to  be  complex  and 

let 

c  =  h-\-  ik, 

wherein  now  h  and  k  denote  two  arbitrarily  assigned  real 
values.     Then,  if  we  let 

z  =  r(cos  ^  +  *  sin  ^), 

r  becomes  infinitesimal  for  every  way  of  approach  of  z  to  the 
origin,  while  the  angle  <^,  made  by  r  with  the  avaxis,  indi- 
cates the  direction  in  which  we  approach  the  origin.  We  now 
obtain 

I  ^  ^^co,.-.•.n.)  ^  ^^L,3/si^\_  ,.  sin(^^)]  ; 

and  if  this  shall  equal  h  +  ik,  the  equations 

C0S(1>  y_i__     i\  COSife 

sin 


e—^o^(^^\  =  h,   e—sm(^^  =  -k, 


cos  <fr  r     •  I  \  7 

or  e  -   =  VFT^   tan(^-^j  =-^ 

must  be  satisfied.     Now —,  for  a  vanishing  value  of  r,  can 

r 

fail  to  be  infinite,  only  when  (f>  simultaneously  approaches  the 
angle  - .  therefore,   if  we   introduce   instead  of  ^  the  angle 

i/f  =  ^  —  «^,  which  r  makes  with  the  y-axis,  and  denote  by  a 

Jj  

the  real  value  of  log  VA'*  -f-  A^,  so  that  a  is  arbitrarily  assumed 


130  THEORY  OF  FUNCTIONS. 

just  as  h  and  k  are,  we  have  instead  of  the  former  equations  to 
satisfy  the  following 

(1)  -J=„,  tau(^-jej=--. 

But  the  former  will  be  satisfied  at  once,  by  letting  ij/  and  r 
tend  towards  zero  simultaneously  in  such  a  way  that 

(2)  i/r  =  ar 

always,  i.e.,  by  letting  the  point  z  approach  the  origin  along 
the  spiral  of  Archimedes  which  is  explicitly  determined  by  the 
value  a,  and  which  is  tangent  to  the  ?/-axis  at  the  origin. 

With  this  relation  existing  between  ij/  and  r,  ^  now  be- 

comes  infinite  as  r  decreases  indefinitely,  and  therefore  -the-  'l^(^ 
tang^it-to  this  curve  is  capable  of  assuming  every  value.     But 

if  we  denote  by  a  the   definite   arc   contained  between  — - 

and  -,  the  tangent  of  which  has  the  value ,  so  that  the 

^  ft 

arbitrarily  assumed  values  h  and  Ti  can  be  replaced  by  the 
equally  arbitrary  quantities  a  and  a,  then  also 

Tc 

tan  {a  +  inr)  =  —  -, 
h 

n  denoting  a  positive  integer.  The  second  of  equations  (1)  is 
satisfied,  therefore,  if  we  assume 

COSi/' 

and  make  r  tend  towards  zero  by  increasing  n  indefinitely.     If 

we  substitute  i-  for  r  conformably  with  equation  (2),  we  get 
a 

a  cos  il/ 

^      a  +  nir 

for  which,  since  cos  \p  differs  from  1  only  by  an  infinitesimal 
of  the  second  order  when  j/^  and  r  are  infinitesimals  of  the 
first  order,  we  can  also  write 

(3)  V'  =  -^- 

a  +  nir 


IXF.  AND  INF'L    VALUES   OF  FUNCTIONS.         131 

1 
Therefore  e*  acquires  the  assigned  value  c  =  h-{-  ik,  if  the 

point  z  approach  the  origin  along  the  spiral   of  Archimedes 

ij/  =  ar  in  such  a  way  that  the  radius  vector  rotates  towards 

the  ?/-axis  per  saltum,  while  the  angle  which  it  makes  with  this 

axis  is  given  by  the  fraction  (3),  of  which  the  numerator  is 

constantly  equal  to  a  and  the  denominator  increases  by  it  with 

every  spring. 

28.  We  shall  now  show  that  a  uniform  function,  which  is 
not  a  mere  constant,  must  become  infinite  at  some  point  z, 
by  proving  the  following  proposition :  If  a  uniform  function  do 
not  become  infinite  for  some  finite  or  infinite  value  of  the  variable, 
it  is  a  constant.  We  can  in  this  case  suppose  the  whole  infinite 
extent  of  the  plane  to  be  the  domain  of  the  origin  and  by  (7), 
§  25,  assuming  a  =  0,  put 

(1)  0(0  =  po  +  p,t  +  pj?  +p^i?  +  ..., 

wherein  »„  =  — -  (  ^^ — . 

We  can,  moreover,  enlarge  indefinitely  the  circle  round  the 
origin,  to  which  this  integral  refers,  without  changing  the 
value  of  the  integral,  since  a  point  of  discontinuity  nowhere 
occurs.     But  if  we  let 


z  =  r(cos  6  -\-isva.  6), 

and  thus 

dz         .-in 

z 

we  get 

2  ttJo      z" 

and  if  we  let  all  the  values  of  z  along  the  circumference  of  the 
circle  become  infinite,  then  p„  vanishes  for  every  value  of  n, 
with  the  exception  of  n  =  0,  since  by  the  hypothesis  <j>(z) 
remains  finite  round  the  circumference  of  the  infinitely  great 
circle.     It  therefore  follows  that 

Pl=P2=P3='-  =0, 


132  THEORY  OF  FUNCTIONS. 

9,nd  series  (1)  reduces  to  its  first  term  Pq,  so  that  the  function 
acquires  the  constant  value 


1    r^" 

L  TTi/O 


for  every  value  of  t. 

We  can  base  the  proof  of  this  proposition  also  upon  equation 
(1),  §  24,  namely, 

^^'        "llziJ      Z-t 

For,  if  we  take  this  integral  along  a  circle  described  round  the 
origin,  we  can  enlarge  that  indefinitely,  because  of  the  assumed 
properties  of  the  function  ^{t).     Accordingly,  if  we  let 

dz        .,n 

—  =  idd, 
z 

we  obtain      ^(t)=^  f  '±^M  =  ^  C' ±^ae. 

^  ^         2  IT  Jo         Z  —  t  2  IT  Jo        ^  t 


If  now  the  radius  of  the  circle  increase  indefinitely,  all  the 
values  of  z  in  the  integral  will  tend  towards  infinity ;  hence 

-  vanishes,  and  the  integral  reduces  to  the  above  constant  value 
z 


1   r'^'^ 


d6 


independent  of  t. 

From  this  proposition  follows  immediately :  If  a  uniform 
function  he  not  a  constant,  it  must  become  infinite  for  some  finite 
or  infinite  value  of  the  variable. 

Further  follows :  A  uniform  function  must  assume  the  value 
zero  for  some  value  of  the  variable.     For,  if  <^  (2;)  be  nowhere 

equal  to  zero, is  nowhere  infinite ;  therefore  — -—  would 

be  a  constant,  and  hence  also  <^  {z). 

Finally :  A  uniform  function  must  be  capable  of  assuming  any 
arbitrary  value  k  at  least  once.  For,  were  <f}  (z)  nowhere  equal 
to  k,  <j>(z)  —  k  would  nowhere  be  equal  to  zero ;  therefore  it 
would  be  constant,  and  so  too  would  <^  (z). 


INF.  AND  INF^L    VALUES   OF  FUNCTIONS.         133 

It  should  be  emphatically  stated  that  these  propositions  no 
longer  hold  absolutely,  if  complex  values  of  the  variable 
be  excluded.  If  only  real  values  be  considered,  the  uniform 
function  cos  z,  for  instance,  does  not  become  infinite  and  does 
not  assume  every  arbitrary  value,  but  only  the  values  between 
—  1  and  4- 1.  Hence  there  exists  here  a  certain  analogy  to 
algebraic  equations,  in  which  also  the  fundamental  proposi- 
tion, that  every  algebraic  equation  must  have  at  least  one 
root,  and  that  every  equation  of  the  nth  degree  has  n  roots, 
is  not  generally  valid  when  only  real  values  are  considered. 

29.  We  turn  now  to  the  consideration  of  the  cases  in  which, 
for  the  function  <l>(z),  the  product  (z  —  a)<f>(z)  no  longer  van- 
ishes at  the  point  z  =  a.  Then  ^(2;)  by  §  27  suffers  here  a 
discontinuity.  Two  possibilities  now  present  themselves: 
either  there  is  a  power  ot  z  —  a  with  a  positive,  integral  or 
fractional  exponent  fi,  for  which  the  product 

(z  —  aY<f>(z) 

has  a  determinate  finite  limit,  or  there  is  no  such  power. 

We  shall  first  consider  the  former  case.  If  this  occur,  let 
us  denote  by  n  the  greatest  integer  contained  in  fi,  so  that 

n -^  fj.  <C  n  -\- 1, 

where  the  equality  holds  when  fi  itself  is  an  integer.     We 
then  have 

lim  [(2-a)"+V(^)]^=a=liai  [(z-ay+'^-''(z-ay<l)(z)],^^=0, 

because  w  +  1  —  /x  is  positive.      But  if  we  divide  by  z  —  a,  J 
then,  according  to  §  27,  p.  128,  I 

(z-a)»</,(2) 

is  a  function  which  remains  finite  for  z  =  a.     If  we  denote  by 
c^"^  the  finite  limiting  value  of  the  same  for  z  =  a,  then 

(z  — a)»<^(2;)— c<") 


134 


THEORY  OF  FUNCTIONS. 


is  a  function  which  vanishes  for  z  =  a,  and  therefore  by  §  27 

C(n) 


(z-ay-'<l>(z). 


z  —  a 


remains  finite  for  z  =  a.      Then,  if  we  denote  by  c^"~^'  the 
finite  limiting  value  of  the  same, 


(z-ay-'cl>{z)-- 
z 

vanishes  for  z  =  a,  and  therefore 
(z  _  ay-^<f>{z)  - 


„(n) 


-  fiC-l) 


r.(n) 


-.(n-1) 


(z  —  ay     z  —  a 

remains  finite  at  the  place  z  =  a.      If  we   continue  in  this 
manner,  we  finally  arrive  at  a  function 


<l>(z)- 


(z  —  a)"     (z  —  ay  -^     (z  -  ay- 


{z  —  ay     z  —  a' 


which  is  finite,  and  hence  also  continuous,  for  z  =  a.     There- 
fore, if  we  let 


z  —  a     (z  —  a)^      (2  —  ay 


{z  —  ay 


\p{z)  denotes  a  function  which  is  finite  and  continuous  for 
z  =  a\  and  if  for  brevity  we  let 


+ 


;  + 


z  —  a  '  {z  —  ay  '  {z  —  ay 
we  obtain 
(2)  4>{z)=A-\-^i,{z), 

wherein 

c(»)  =  lim  [(2  -  ay<i>{z)\^^, 


+  •••  + 


r.(») 


(2  -  a)" 


=  A, 


g(n-l)  _  lij^ 
g(n-2)  _  Jjj^ 


(2-a)"-V(2!)- 


etc. 


r.(») 


r.(»-l)" 


(2!  —  a)^     1?  —  a_ 


INF.  AND  INF'L    VALUES  OF  FUNCTIONS. 


136 


By  this  means  a  part  A,  which  becomes  infinite  only  for  z  =  a, 
is  separated  from  <i>(z),  the  additional  part  ij/(z)  remaining 
finite  for  z  =  a.     Now  if  the  finite  constant  c^"^  do  not  have 


the  value  zero,  i.e.,  if  the  term 


r.(«) 


be  not  wanting  in  the 


(z  —  a)" 

expression  A,  we  can  say :  If  Urn  [(2  —  a)"^(z)]^ia  he  neither 
zero  nor  infinite,  the  function  <l>(z)  is  infinite  of  the  nth  order  for 
z  =  a.  In  that  case,  however,  this  condition  is  not  satisfied 
for  any  fractional  exponent  fi,  but  lim  (z  —  a)i^(f>(z)  is  either 
zero  or  infinite ;  for,  if  /u.  >  n,  as  we  originally  assumed,  then 

lim  [(z  -  aY<i>{z)\^  =  lim  [(z  -  ay-\z  -  ay<f>(z)^^  =  0, 

but  if  /A  <  w,  then 


lim  [(z  -  aycf,(z);\,^^  =  lim 


(z  -  aY<f>(zy 
_  (z  —  a)"  '^ 


Therefore  <f>{z)  cannot  be  infinite  of  a  fractional  order,  and 
the  proposition  follows  :  If  a  uniform  function  become  infinite 
of  a  finite  order,  it  can  be  infinite  only  of  an  integral  order. 

An  example  may  be  added  to  the  preceding  theory.     The 
function 

z^{z  —  ly 

is  uniform  and  has  the  points  of  discontinuity  z  =  0  and  z  =  1. 
For  z  =  0  we  have 

1 


c"'  =  lim[z3<^(z)],^  =  lim 


i{z-iy 


=  1 


therefore  c'"  is  finite  and  not  zero,  and  hence  ^(z)  is  infinite  of 
the  third  order  for  z  =  0.     Now  since 


lim 


l]     =0, 


L(^-i) 

we  obtain  after  dividing  by  z  the  finite  value 

1 1' 

_z(z  —  iy    z 


c"  =  lim 


=  2. 


136  THEORY  OF  FUNCTIONS. 

In  like  maimer 


and  finally 


3'  =  lim  — 

\_z\z  -  ly     z' 


=  3, 


^(z  -  ly 


hy^^= 


3\      4-3^  . 

zj     (z  -  ly ' 


accordingly  the  separation  into  the  two  parts  A  and  \j/(z)  is 
the  following : 


*(«)  = 


\Z       Z^        ^ 


4-32 


2^(2  -  1)2        ^2        2^        2^;   '    (2  -  1)2 

For  the  other  point  of  discontinuity,  2  =  1,  we  have 


lim   (z-iy<l>(z) 


..rHi]  ='' 


and  therefore  </>(2)  is  infinite  of  the  second  order  for  2  =  1. 
After  division  by  2  —  1  we  obtain 


;' =  lim =  lim    — 

[2^(2-1)  (2-i)j,,,      L 


and  then 


2^  +  2+11 

^            J 

^1 

32^  +  22  +  1 

=  -3, 


2^(2  -  ly       I   (2  -  1)2        (Z-1)\  2^ 

Therefore  the  separation  in  this  case  is  the  following ; 


<p(z)  = 


2^(2  -  1)2 


+ 


(2-l)2_ 


+ 


322  ^2z+l 


In  the  cases  considered,  where  ^{z)  becomes  infinite  of  the 
nth  order  for  z  =  a,  the  discontinuity  is  always  a  polar;  for  if 
we  let 

(z-ay<l>(z)  =  F{z), 

F(z)  assumes  a  definite  finite  value  different  from  zero  for  all 
paths  of  approach  to  a,  and  therefore 

1    =(^  — «)" 
<^(2)  F{z) 


INF.  AND  INF'L   VALUES  OF  FUNCTIONS.        137 

acqiiires  the  value  zero  for  all  paths  of  approach.  Conse- 
quently ^(2;)  suffers  a  discontinuity  of  the  first  kind  (p.  126). 
From  this  it  follows  further  that,  when  <ji(z)  is  infinite  of  the 
Tith  order  for  z  =  a,  we  can  let 


<l>{z)  = 


F(z)_ 


{z  -  a)»' 

wherein  F{z),  for  2  =  a,  is  finite  and  not  zero,  and  conversely. 
This  form,  which  we  can  give  the  function  <ji(z)  in  the  case 
considered,  warrants  the  assumption  that  an  infinity  of  the  «th 
order  can  be  looked  upon  as  a  coincidence  of  n  points,  at  each 
of  which  <^{z)  is  infinite  of  the  first  order,  or  as  an  infinity  of 
multiplicity  n.  For,  if  <^(2;)  become  infinite  of  the  first  order 
at  two  points  a  and  h,  say,  we  can  conformably  with  the  above 
principles  let 

(z-a) 

wherein  F{z),  for  z  =  a,  is  not  infinite,  but  is  so  for  2  =  6,  and 
that  of  the  first  order.     Therefore  we  have  further, 

2  —  0  (z  —  a){z  —  b) 

wherein  Fi{z)  is  not  infinite  or  zero  at  a  or  at  b.     Now  if  the 
points  a  and  b  coincide,  it  follows  that 

(2  —  ay 

and  therefore  <f>{z),  by  the  above  criterion,  is  infinite  of  the 
second  order  at  a. 

We  saw  above  that,  when  a  function  <f>(z)  is  infinite  of  a 
finite  order  for  z  =  a,  it  suffers  here  a  discontinuity  of  the  first 
kind;  we  will  now  show  that  the  converse  is  also  true.     If 

<f>{z)  have  a  polar  discontinuity  at  the  point  z  =  a,  then  — — 

<l>(z) 

is  continuous,  and  has  the  value  zero  at  this  place.     We  can, 
therefore,  by  (7),  §  25,  let 

(3)     T7-X  =Pi(^-  «)  +  i>2(2'  -  af  +  ...  +p„(2  -  ay  +  ... ; 
<f>{z) 


138  THEORY  OF  FUSCTIONS. 

for  the  first  term  p^  must  be  wanting,  since  it  has  the  value 
acquired  by  — —  at  z  =  a,  and  this  is  zero.     Of  the  following 

coefficients,  some  may  also  be  zero.  Let  the  first  which  does 
not  vanish  be  p^.  Such  a  coefficient  must  exist,  otherwise 
— --  would  be  constant,  and  would  have  the  value  zero  for 
every  value  of  z.     Therefore  let 

T7T  =  i>«  («  -  a)"  +  Pn+\{^  -  a)"^'  4-  •  •  •, 

wherein  p^  is  finite  and  different  from  zero.  Now  if  we  bring 
this  to  the  form 

— -  =  (z  _  a)"  [i>,  +i),+i (2  _  a)  +  ...] 
and  let  ^ =  F(z), 

we  have  ^(z)  =  ,    ^^'    : 

but  since  F(z),  for  z  =  a,  acquires  the  value  — ,  finite  and 

different  from  zero,  then  ^(z)  becomes  by  the  above  criterion 
infinite  of  the  ?ith  order,  and  therefore  of  a  finite  order.  ^ 

Consequently  the  occurrence  of  a  polar  discontinuity  at  a 
point  a  is  always  characterized  by  the  property  that  the 
function  becomes  infinite  of  a  finite  order  at  that  point. 

From  this  it  follows  at  once  that  the  case  mentioned  on 

^^^^.      P- 126  (note),  that  <^(z)/always  becomes  infinite  at  a  point  a  for 

different  paths  of  approach  to  this  point,  but  infinite  of  differ- 

!|r        ent  orders,\is  in  fact  not  possible,  but  introduces  a  contradiction. 

In  that  case  — -—  would  receive  the  value  zero  for  all  paths  of 


vtr^nrr  .         ..    ^     approach  to  a.    But,  as  was  shown  above,  <f>(z)  becomes  infinite 
\jS4^    first  in  (3)  not  to  vanish. 


^fljy^'^*^^'^  J      of  ^  definite  order  determined  by  that  coefficient  which  is  the 


fi^jh^l'^  '^ 


1  Konigsberger,  Vorlestmgen  ilberdie  Theorie  der  ell.  Funkt.,  I.  S.  121. 


INF.  AND  INF'L    VALUES  OF  FUNCTIONS.         139 

We  now  turn,  our  attention  to  the  second  possibility  men- 
tioned on  p.  133,  namely,  that  there  is  no  power  of  z  —  a  with 
a  finite,  positive  exponent  /x,  for  which  the  product  {z—aY<l>(z) 
acquires  a  finite  value  for  all  paths  of  approach  to  a.  Accord- 
ing to  the  preceding,  this  can  occur  ^nly  in  the  case  of  a  dis- 
continuity of  the  second  kind.  But  the  series  derived  (10), 
§  26,  holds  for  the  latter,  because  for  that  development  ihe 
discontinuity  occurring  at  a  could  be  an  entirely  arbitrary  one, 
the  point  a  having  been  excluded  by  means  of  a  small  circle  C. 

If  in  (10),  §  26,  we  let 

Po  -\-Pi(z-a)+  p^  (z  -  a)2  +  ...  =  if/(z), 

so  that  (/'(z)  represents  a  finite  and  continuous  function  for 

2  =  a,  we  obtain 

pi  pii  pill 

(4)  ^(z)=.^  +  -^+-^+...+^{z). 

z  —  a     (z  —  ay     {z  —  ay 

In  this,  by  (9),  §  26, 

c(n+n  =    1    C^(z)(z  -  aYdz, 

2  7rt'»/ 

the  integral  being  taken  along  the  circle  C  described  round  a. 

If  we  substitute  in  that  integral 

dz 

z  —  a  =  r(cos  6  -\-ism  6),    =  idO, 

z  —  a  . 

we  have  c("+i>  =  ^  C'^ <i>{z){z  -  ay+^dS. 

Now  if,  in  order  in  the  first  place  to  consider  the  former 
case  from  this  point  of  view,  <f)(z)  be  infinite  of  the  7ith  order 
for  z  =  a,  then  (z  —  ay<f)(z)  is  finite  at  a,  and  therefore 
(z  —  ay'^^<t>(z)  is  zero.  Hence,  if  the  radius  of  the  circle  C  be 
made  to  tend  towards  zero,  c^"+^>  and  with  greater  reason  all 
succeeding  coefficients,  c<"+->,  c<"+'',  ...,  vanish.      Therefore  the 

series  contained  in  (4)  ends  with  the  term and  changes 

(z  —  ay 

into  the  expression  A,  found  previously  under  (1).  If,  on  the 
contrary,  the  second  possibility  already  mentioned  occur,  in 
which  (z  —  a)"<^(z)  does  not  have  a  finite  limit  for  any  finite 


140  THEORY  OF  FUNCTIONS. 

value  of  n,  then  none  of  the  coefficients  c^">  vanish,  and 
the  infinite  series  contained  in  (4)  enters  in  place  of  the  former 
expression  A.  In  this  case  <^{z)  is  infinite  of  an  infinitely  high 
order  for  z  =  a,  and  at  the  same  time,  as  remarked  above,  the 
discontinuity  at  a  is  of  the  second  kind. 

Therefore  the  two  kinds  of  discontinuity  are  also  character- 
ized by  this,  that  in  the  first  occurs  an  infinity  of  a  finite 
order,  in  the  second  one  of  an  infinitely  high  order. 

We  now  return  to  equation  (2), 

<i>{z)=A  +  rp{z), 

in  which  A  denotes  either  the  finite  series  (1) 

z  —  a      (z  —  ay  (z  —  ay 

or  by  (4)  an  infinite  series  of  the  same  form ;  (i/^)z,  however, 
denoting  a  finite  and  continuous  function  at  a.  This  equation 
shows  that  a  uniform  function  <i>{z),  which  becomes  infinite  at 
a  place  a,  is  distinguished  from  a  function  ^(z),  which  remains 
finite  there,  only  by  an  expression  of  the  form  A.  Hence  it 
becomes  infinite  only  as  this  expression  A  does.  For  exam- 
ple, if  ^{z)  be  infinite  of  the  first  order  for  z  =  a,  so  that 
lim  [(z— a)^(2;)]^a  is  finite  and  not  zero,  we  can  then  also  say  that 

<j>{z)  becomes  infinite  there  just  as  does.     Or,  if  <j){z)  be 

infinite  of  the  second  order  for  z  =  a,it  is  then  infinite  either 

c'  c"  c" 

as h -,  or  only  as is.     If  we  have  another 

z  —  a      (z  —  ay  (z  —  ay 

uniform  function  f{z),  which  likewise  becomes  infinite  of  the 
nth.  order  for  z  =  a,  this  can  also  become  infinite  only  as  a 
similar  expression  A  does,  which  can  differ  from  the  former 
only  in  the  value  of  the  coefficients  c.  If  the  latter  function 
f(z)  be  given,  the  coefficients  c  are  thereby  also  given ;  there- 
fore <}>(z)  is  known  at  a  place  of  discontinuity  a,  if  a  function 
/(z)  be  given,  which  becomes  infinite  at  this  place  just  as 
<f}(z)  does.     We  can  then  let 

<f>iz)  =  f(z)  +  il^{z), 
wherein  ^  (z),  for  z  =  a,  remains  finite  and  continuous. 


INF.  AND  INF'L    VALUES   OF  FUNCTIONS.         141 
From  the  equation     <fi(z)  =  A-\-  x^iiz) 
follows  by  differentiation 

where 

dA  c'  2  c"  3  c'"  nc("> 


dz  (z-af      {z-af      (z  -  a)*  (z  -  a)"+i 

Now  since  (by  §  24)  \l/'(z)  remains  finite  for  z  =  a,  because 
tl/(z)  is  here  finite,  we  have :  The  derivative  <f>'(z)  of  a  uniform 
Junction  (f>{z)  at  a  pla/^e  z  =  a,  where  <l>(z)  is  infinite,  becomes 
likewise  infinite,  and  that  of  an  order  higher  by  unity  than 
<f>(z).  At  all  finite  points  at  which  <f)(z)  is  finite,  4>'(z)  also 
remains  finite  (by  §  24),  and  hence  the  finite  points  of  discon- 
tinuity of  a  uniform  function  are  identical  with  those  of  its 
derivative  <t>'{z). 

30.  We  now  proceed  to  the  inquiry,  how  a  uniform  func- 
tion <f)(z)  behaves  for  an  infinite  value  of  the  variable  z.  We 
can  lead  this  investigation  back  to  the  preceding  by  putting 

z  =  -,  whereby  <ji(z)  may  change  into  f(u),  and  then  examin- 

u 
ing  f(u)  at  the  point  u  =  0.     Now,  in  the  first  place,  f(u)  is 
finite  for  M  =  0  (by  §  27)  when  lim  [t</(w)]„^o  =  0.     Therefore 


<f>(z)  is  finite  for  z  =  cc  when  lim 


-<f>(zy 


0. 


Further,  f(u)  becomes  infinite  of  the  nth.  order  or  of  multi- 
plicity n  (by  §  29)  when  lim  [w"/(m)],^  is  neither  zero  nor 
infinite.  , 

Hence    <f){z')  is  infinite  of  the  nth  order  for  z  =  cc  when 
Um\^~ 
is  neither  zero  nor  infinite. 


142  THEORY  OF  FUNCTIONS. 

Moreover,  we  can  (by  §  29)  in  this  case  put 

Qt        nil        QUI  Qin) 

where  X  (u)  denotes  a  function  which  remains  finite  for  w  =  0, 
and  the  quantities  Q  constant  coefficients.  If  \  («),.  expressed 
in  terms  of  z,  change  into  tp  (z),  we  obtain  from  the  preceding 
equation  the  following : 

(1)  <l>{z)=  Q'(2)+  Q"z'  +  Q'"!^  +'•'  +  Q<"'z-  +  ^(z), 

wherein  ipiz)  remains  finite  for  z  —  zc.      In  this  case,  there- 
fore, <f>(z)  is  infinite  just  as  an  integral  function  of  z  is. 
From  equation  (1)  follows 

(2)  <f>'(z)=  Q'  +  2  Q"z  +  3  Q"'zF  +  ...  +  »iQ(">z-^  +  il>'{z). 

To  inquire,  in  the  first  place,  how  the  derivative  il/'(z)  of  the 
function  il/(z)  (which  remains  finite  at  infinity)  behaves  at 
that  point,  we  introduce  again  the  variable  u.     Since 

and  \j/(z)  =  X(m), 

we  have  ^'(z)  =  —  m*X'(«). 

Now  A(m)  is  finit-e  for  «  =  0,  therefore  by  §  24  X'(«)  is  also 
finite,  and  consequently 

^'(z)  =  0  for  z  =  00. 

Therefore,  if  a  uniform  function  be  finite  at  the  point  z  =  oo, 
its  derivative  is  equal  to  zero  at  that  point.     For  example, 

z'  +  z  +  l 
2z2-l  ' 

Then  it  follows  from  (2)  that  <f>'(z)  is  infinite  of  an  order 
less  by  unity  than  <f>(z),  at  z  =  oc.  Therefore,  if  <^(z)  be  infi- 
nite of  the  first  order  only,  <f>'(z)  remains  finite  for  z  =  oo. 


INF.  AND  INF'L   VALUES  OF  FUNCTIONS.         143 

The  integral  function  of  z  occurring  in  (1)  can  be  derived 
also  from  the  series  obtained  §  26  (13),  which  holds  when 
^(z)  suffers  a  discontinuity  of  any  kind  at  z  =  oo ;  it  is  valid 
then  for  all  points  z  lying  outside  a  circle  which  encloses  all 
finite  points  of  discontinuity.  If  we  denote  by  ^(z)  the  first 
part  of  that  series  and  put 

this  function  remains  finite  for  z  =  »  and  assumes  the  value 
Pff  Denoting  the  other  coefiicients  by  Q  instead  of  by  c,  we 
therefore  obtain  from  (13),  §  26, 

(3)  <^(2)=  q'z  +  Q"z*  4-  Q"'^  +'•'+  ^{z\ 

wherein  by  (12),  §  26, 

the  integral  to  be  taken  along  a  circle  round  the  origin,  outside 

which  there  is  no  point  of  discontinuity  except  z  =  «.    By 

dz 
substituting  therein  —  =  idd,  we  obtain 

z 


1    r 


"^(Z 


dd. 

z" 


Now  if  </)(z)  be  infinite  of  the  «th  order  for  z  =  oo,  then 
lim  I  ^i5l         is    finite,   and   therefore    lim    ^^         is    zero. 

Consequently,  if  we  let  the  circle  of  integration  enlarge  indefi- 
nitely, Q<"+*^  Q^"'*'^,  etc.,  vanish,  and  the  series  contained  in 
(3)  changes  into  the  integral  function  in  (1). 

But  when  <f>(z)  is  infinite  of  an  infinitely  high  order,  and 
when  therefore  it  suffers  a  discontinuity  of  the  second  kind, 
then  in  place  of  the  integral  function  in  (1)  there  enters  the 
series  of  integral  ascending  powers  of  z  in  (3). 

31.  From  the  preceding  investigation  we  now  deduce  the 
following  propositions:  If  a  uniform  function  become  infinite 


144  THEORY  OF  FUNCTIONS. 

for  no  finite  value  ofz,  hiit  only  for  z  =  cci,  and  that,  only  of  a 
finite  order  (multiplicity  n),  then  it  is  an  integral  function  of 
the  nth  degree.     For  we  have  in  this  case  by  (1),  §  30, 

^(z)=  Q'z  +  Q"z'  +  Q'"^  +  -  +  Q("V  +  il^iz). 

But  since  il/(z)  is  a  uniform  function,  which  does  not  become 
infinite  either  for  a  finite  or  for  an  infinite  value  of  z,  it  is  by 
§  28  a  constant.     Denoting  it  by  Q,  we  have 

<^(z)=  Q+Q'z+  Q"z'  +  Q'"^  +  ...  +  Q(»)2"; 

thus  cf>(z)  is  in  fact  an  integral  function  of  the  ?ith  degree. 
Conversely,  an  integral  function  of  the  nth  degree  becomes 
infinite  only  for  z  =  cc,  and  that  of  multiplicity  n ;  for 


lim 


z" 


=  Q 


i(n) 


and  this  limit  is  finite  and  at  the  same  time  different  from  zero, 
when  <f)(z)  is  of  a  degree  not  less  than  the  nth. 

If  a  uniform  function  ^(z)  become  infinite  only  for  z  =  cxi,  but 
that  of  an  infinitely  high  order,  then  it  can  be  developed  in  a  series 
of  powers  ofz  converging  for  every  finite  value  ofz.  For  in  this 
case  the  series  (3),  §  30,  holds  for  all  finite  values  of  z,  and  1/^(2;) 
must  be  a  constant  for  the  same  reason  as  before. 

32.  If  a  uniform  function  become  infinite  only  for  a  finite 
number  of  values  of  the  variable,  and  for  each  only  of  a  finite 
order  (in  short,  if  it  become  infinite  only  a  finite  number  of  times), 
then  it  is  a  rational  function. 

Let  a,  b,  c,  .••,  k,  I,  00  be  the  values  of  z  for  which  <j)(z)  becomes 
infinite,  a,  ^,  y,  •••,  K,X,fi,  the  respective  multiplicities  of  the 
infinities;  then  we  can  in  the  first  place  let 

<l>(z)  =  Q'z  +  Q"z'  +  -  +  Q^'^V  +  tl,(z), 

where  \J/(z)  is  not  infinite  for  z  =  00,  and  therefore  is  infinite 
only  for  z  =  a,b,c,"-,l',  accordingly  we  have 


(a) 


z  —  a      (z  —  a)-  (z  —  ay 


INF.  AND  INF'L   VALUES  OF  FUNCTIONS.         145 

where  now  ^^(z)  is  infinite  only  for  z  =  h,c,"',l.  Therefore  we 
have  further  * 

^■W=^  +  (;^  +  ...  +  ^+^,(.). 

If  we  continue  in  this  way,  we  arrive  at 

^»-(^)=^  +  (£0^  +  -  +  ^  +«=>). 

where  ipjiz)  is  no  longer  infinite  at  all  and  therefore  is  a  con- 
stant. Denoting  this  by  Q,  we  obtain,  when  we  combine  the 
above  expressions, 

<i>{z)  =  Q  +  Q'z  +  q'z"  +  ■  •  •  +  Q^'^V 


z  —  a      {z  —  df  (z  —  a)" 


+ 


+  ^^,  +-7-^~r.  +•••  + 


(A) 


z-i^  (z-iy        ^  (z-i)^' 

hence  <^(z)  is  in  fact  a  rational  function. 

Since  a  rational  function  can  always  be  brought  to  the  above 
form,  that  is,  can  be  separated  into  an  integral  function  and 
partial  fractions,  it  follows  also,  conversely,  that  a  rational 
function  can  always  become  infinite  only  a  finite  number  of 
times. 

33.  A  uniform  function  <^{z)  is  determined,  except  as  to  an 
additive  constant,  when  for  each  of  its  points  of  discontinuity  we 
are  given  a  function  which  becomes  infinite  at  this  point  just  as 
<f>(z)  does,  but  which  otherwise  remains  finite  and  continuous. 

Let  ttj,  tta,  Og,  etc.,  be  the  points  of  discontinuity  of  ^{z),  and 
suppose  the  value  00  to  be  included  among  them.  Further,  let 
fii^)}  f 2(^)9  fsi^))  etc.,  be  the   given  functions,  which  become 


146  THEORY  OF  FUNCTIONS. 

infinite  at  the  points  ai,  a^,  a^,  etc.,  respectively,  but  which  are 
elsewhere  finite  and  continuous.  Then,  if  ^(z)  is  to  become 
infinite  at  Oj  just  as  fi(z)  does,  we  can  let 

where  i{/(z)  is  not  infinite  for  z  =  a^.  Now,  since  fi(z)  is  finite 
for  2  =  ttg,  \p{z)  must  be  infinite  at  that  point,  and  that  just  as 
<f)(z)  is.  Hence,  if  <f>(z)  is  to  be  infinite  at  aj  just  as  /2(z)  is,  we 
can  let 

'/'(«)=/2(z)+«^i(z), 

where  now  x{/i(z)  does  not  become  infinite  for  ai  and  aj,  but  does 
for  ttg,  etc.  If  we  continue  in  this  way,  we  finally  arrive  at  a 
function  if/  which  is  a  constant,  since  it  no  longer  becomes 
infinite  at  any  point.     Denoting  this  constant  by  C,  we  obtain 

<l>(z)=Mz)-\-Mz)+Mz)+ ...  +  a 

34.  We  say  that  a  uniform  function  <t>(z)  becomes  infinitesi- 
mal or  zero  of  the  nth  order  for  a  value  of  z,  when  becomes 

infinite  of  the  nth  order  for  this  value.  For  this  case,  by  §  29 
and  §  30, 


limf. 


"<^(^)i. 


is  neither  zero  nor  infinite. 


Now,  since  the  reciprocal  fractions  must  also  have  finite  limits 
different  from  zero,  we  have  as  the  conditions  to  ensure  that 
0(z)  is  infinitesimal  or  zero  of  the  nth  order  for  a  finite  value 
2  =  a,  and  for  z  =  cc: 


l{z-ay 
lim[2»<^(2)]^ 


is  neither  zero  nor  infinite. 


From  these  conditions  are  derived  the  former  ones  for  the  infi- 
nite state  of  <f>(z)  by  substituting  —  n  for  n ;  hence  we  can 


INF.  AND  INF'L   VALUES  OF  FUNCTIONS.        147 

consider  an  infinite  value  as  an  infinitesimal  value  of  a  nega- 
tive order,  or  also  conversely. 

If  <f>{z)  become  zero  of  the  nth  order  for  z  =  a,  and  if  we  let 


(2  —  a) 


then  according  to  the  above  F(z)  is  a  function  which  is  finite 
and  not  zero  for  z  =  a.     From  this  it  follows  that  we  can  let 

cfy(z)  =  (z-  ayFiz), 

and  therefore  remove  the  factor  (z  —  a)"  from  <^  (z).  If  we 
replace  w  by  —n,  we  obtain  again  the  condition  given  on  p.  137, 
that,  if  ^(z)  be  infinite  of  the  nth.  order  for  z  =  a,  we  can  let 


<^(z) 


ZM. 


(2  -  a)»' 
and  conversely. 

If  <f>(z)  become  infinitesimal  of  the  nth   order  for  z=ao, 

then  z"<l>{z)  =  F  (z) 

is  finite  and  not  zero  for  z  =  00 ;  and  this  equation  holds  also 
for  infinite  values,  if  —  «  be  substituted  for  n.  Hence  in  this 
case,  for  infinitesimal  values  of  ^(z),  we  can  let 

^  ^        z" 
and  for  infinite  values 

<t>(z)  =  z''F(z), 

wherein  F(z)  denotes  a  function  which  remains  finite  and  not 
zero  for  2;  =  oc . 

35.   Closely  associated  with  the  preceding  is  the  inquiry, 

how  often  in   a  given  region   a  uniform   function  becomes 

infinite  i-  ^^     r-     ,       ^       ■       -,  ■  -,         infinite  , 

.  „  .^  .  ,  of  the  first  order,  m  which  an  .  _  .,  .  ,  value 
infinitesimal  innnitesimal 

of  the  nth  order  is  regarded  as  an  .    „   .^     .      ,  value  of  the 

infinitesimal 


148  THEORY  OF  FUNCTIONS. 

first  order  of  multiplicity  n.     This  number  can  be  expressed 

by  a  definite  integral.^    Within  a  given  region  T  let  the  uni- 

,  ^  ,  infinite  ,  ,, 

form  function  ^(z)  become  .  ^   .,     .      ,  at  the  points  a^,  a^,  a^, 

etc.,  of  orders  «i,  n^,  %,  etc.,  respectively,  which  are  to  be  taken 
positively  for  infinitesimal,  negatively  for  infinite  values.  We 
will  now  consider  the  integral 


/dlog<^(.)or/^).., 


taken  over  the  whole  boundary  of  T.     The  function  xAJ  be- 

comes  infinite  at  all  points  at  which  ^(2;)  is  zero,^  and  also  at 
all  points  at  which  ^'  (z)  is  infinite.^  But  by  §  24,  <^'  (z)  remains 
finite  at  all  points  at  which  ^(z)  is  finite,  and  by  §  29  becomes 
infinite  at  all  points  at  which  <^{z)  is  infinite ;  hence  the  points 
of  discontinuity  of  <^'(^)  within  T  are  identical  with  those  of 

^(2).     Accordingly  _z_^   becomes  infinite  at  all  the   points 
<f>{z) 

Oi,  ttj,  ttg,  etc.,  and  only  at  these.  Now  by  §  19  the  above  inte- 
gral, taken  along  the  boundary  of  T,  is  equal  to  the  sum  of  the 
integrals  taken  round  small  circles  described  round  the  points 
a.     Let  A  denote  one  of  these  integrals  corresponding  to  the 

point  a,  and  let  n  denote  the  order  of  the  .   „   .,     .      ,  value 

infinitesimal 

of  <i>{z)  at  that  point.     Then  by  §  34  we  have  * 

<i>{z)  =  {z-ayxl;{z), 
where  ^  (z),  for  z  =  a,  remains  finite  and  different  from  zero. 

1  This  occurs  first  in  Cauchy's  writings,  Comptes  rendus,  Bd.  40,  1855, 
I.,  "M^moire  sur  les  variations  integrals  des  fonctions,"  p.  656. 

2  It  is  evident  from  (t)(s)  =  (z  —  aY^p^z)  (n  being  positive)  that  0'(z) 
is  finite  when  (p{z)  is  infinitesimal  of  the  first  order,  and  in  general  that 
0'(z)  is  infinitesimal  of  an  order  lower  by  unity  than  (t>{z).     [Tr.] 

8  Because  <p'{z)  is  infinite  of  an  order  higher  by  unity  than  0(z) 
(p.  141).     [Tr.] 

*  In  the  relation  given,  n  (as  always  now  in  the  considerations  follow- 
ing) is  to  be  taken  positively  for  infinitesimal,  negatively  for  infinite 
values.     [Tr.] 


INF.  AND  INF'L   VALUES  OF  FUNCTIONS.         149 

By  means  of  this  relation  we  obtain 

the  integral  to  be  taken  along  a  small  circle  described  round 
a.     Since  i{/(z)  is  not  zero  and  i{/'(z)  not  infinite  within  the  circle 

of  integration,  ^  ^  ^  is  continuous,  and  hence  by  §  18 


s 


/dz 
=  2  Trt, 
z  —  a 

and  therefore  A  =  2  irin. 

If  we  sum  up  these  values  for  all  points  a,  we  obtain 

d  log  ^{z)  =  2  7rt(wi  +  ^2  +  «3  +  •  •  •)  =  2  7rt5w, 

the  integral  to  be  taken  along  the  boundary  of  T} 


/< 


1  At  this  point  we  have  omitted  from  the  text  the  following:  Therein 

2n  indicates  how  often  <b(z)  becomes  .   „  .^    .      ,  of  the  first  order,  if 
.«„.,„      ^^  ^  mfinitesimal  ■  g   -4. 

,         mnnite  ,         r  xi.       ^t.       j  infinite 

we  regard  an  .   „   .^    .      ,   value  of  the  nth  order  as  an   .   „   . 

innnitesimal  infimtesimal 

value  of  the  first  order  of  multiplicity  n.    We  therefore  have  the  follow- 
ing proposition :     The  integral 


^d  log  <f,(z) 


of  a  uniform  function  <^(2),  taken  along  the  boundary  of  a  region  T,  is 
equal  to  2  vi  times  the  number  of  points  within   T  at  which  <f>{z)   is 

.    ,   .^    .      ,  of  the  first  order, 
mjimtesimal 

This  statement  of  the  result  is  evidently  misleading,  because  Sn  is  the 

algebraic  sum  of  the  orders  of  the  infinitesimal  values  of  <f>{z)  (the  infinite 

values  being  regarded  as  infinitesimal  values  of  negative  orders,  as  often 

stated).    Thus  if  5  be  the  number  of  points  at  which  4>(z)  becomes  zero, 

and  3  the  number  of  points  at  which  it  becomes  infinite  (account  being 

taken  in  both  cases  of  any  multiplicities),  then  Sn  =  5  —  3  =  2.     The 

statement  is  correct,  however,   if  only  infinitesimal,   or   only  infinite 

values,  be  included  in  the  area  T.     [Tr.] 


150  THEORY  OF  FUNCTIONS. 

If  we  let  the  letter  n  refer  to  the  infinitesimal  values,  and 
denote  by  —  v  the  orders  of  the  infinite  values  (since  these  are 
to  be  taken  negatively),  we  obtain 

(1)  Cd  log  i>(z)  =  2  Trt  (2n  -  2v). 

If  the  function  (f>(z)  remain  finite  within  T,  the  term  —  2v 
drops  out  of  the  preceding  formula,  and  it  follows  that :  The 
number  of  points  at  which  a  uniform  function  <fi(z)  is  zero  of  the 
first  order  within  a  region  T,  in  which  <l>(z)  is  continuous,  is 
equcU  to 

J-.  fd\og<f>{z), 

taken  round  the  boundary  of  T. 

36.  If  we  understand  by  the  points  a  all  Jinite  points  at  which 
<j>(z)  becomes  infinite  or  infinitesimal,  it  is  still  important  to 
determine  the  behavior  of  <f>  (z)  for  z  =  cc.     Let  us  assume  that 

d>(z)  is  .    „   .,     .      ,  of  the  mth  order  for  z  =  cc,  and  again  let 
^^  ^      infinitesimal 

a  positive  m  refer  to  an  infinitesimal  value,  a  negative  m  to  an 

infinite  value.     If  the  boundary  of  T  be  assumed  to  be  a  circle 

roimd  the  origin,  which  encloses  all  the  points  a,  then  in  the 

first  place  according  to  the  preceding 

(1)  Cd  log  <l>{z)  =  2  7rt  (2n  -  2v), 

taken  round  this  circle.     Now,  if  a  new  variable  u  be  intro- 
duced in  place  of  z  by  the  relation 

1 

z  =  -, 
u 

then  to  every  point  z  corresponds  a  point  u,  and  to  the  point 
2  =  CO,  the  point  w  =  0.     Further,  if  we  put 

z  =  r  (cos  6  +  i  sin  6), 

we  get  u  —  -  (cos  6  —  i  sin  B). 

r 


INF.  AND  INF'L   VALUES  OF  FUNCTIONS.        151 

"Wlien  z  describes  a  closed  line  Z  round  the  origin,  u  describes 
likewise  a  closed  line  U  round  the  origin  (because  thereby  6 
increases  from  0  to  2  tt),  but  in  the  opposite  direction.     If  the 

radius  vector  r  be  made  to  increase,  6  remaining  constant,  - 

r 

decreases,  and  conversely ;  therefore  to  all  points  z  outside  Z 
correspond  points  u  situated  within  JJ.  If  we  now  introduce  u 
in  place  of  z  in  the  integral 


faio,^iz)orfmaz, 


<i>(z) 

and  denote  by  f(u)  the  function  thereby  resulting  from  <^(2), 
we  obtain 

/dlog/(»)or/^d». 

In  the  integral  as  to  z,  the  curve  of  integration  Z  is  a  circle 
round  the  origin  enclosing  all  points  a ;  therefore,  in  the  inte- 
gral as  to  u,  the  curve  of  integration  is  also  a  circle  round  the 
origin,  which,  however,  is  described  in  the  opposite  direction. 
Hence,  if  we  assume  for  both  integrals  the  integration  in  the 
direction  of  increasing  angles,  we  have 


id  log  <^{z)  =-Jd iogf(u), 


the  first  integral  taken  round  the  circle  Z,  the  second  round 
the  circle  U.     The  circle  Z  encloses  all  points  a ;  therefore  ^(z) 

becomes  .   ^   ■.     •      i  outside  Z  only  for  z  =cc,  and  hence  f(u) 
infinitesimal  -^  '  •'^  ^ 

mthin  U  is  .  ^   .^    .      ,  only  for  u  =  0.     For  z  =  co,  <f>(z)  was 
infinitesimal        ''  '  -r\  / 

.    -   .      .      ,  of  the  mth  order,  so  that 
infinitesimal 


>-t"«^)L.="-[f^L 


is  finite  and  not  zero ;  accordingly  f(u)  is  also  .    „   .      .      ,  of 
the  mth  order  for  w  =  0.     If  we  let 


162  THEORY  OF  FUNCTIONS. 

tj/(u)  denotes  a  function  which  is  finite  and  different  from 
zero  for  m  =  0,  and  therefore  everywhere  within  U.  Now  it 
follows  as  above  that 


JdiogA«)  =  ™/~+/^<i«, 


wherein  the  second  integral  vanishes,  and  the  first,  taken  in 
the  direction  of  increasing  angles,  is  equal  to  2  irim.  Accord- 
ingly we  have 

I  d  log  ^(z)  =  —  id  log/(?*)  =  —  2  trim. 

If  we  equate  this  result  to  the  value  of  this  integral,  taken 
along  the  same  curve,  found  in  (1),  we  obtain 

(2)  2«  —  Sv  =  —  w. 

If  now  ^(2)  be  zero  for  z  =  oo,  then  m  is  positive,  and  we  have 

m  +  Sw  =  2v ; 

but  if  ^(z)  be  infinite  for  2;  =  00,  then  m  is  negative,  and 
substituting  —  ju,  for  it,  we  obtain 

5n  =  /u,  +  %v. 

In  both  equations,  the  left  side  shows  how  often  ^{z)  becomes 
zero  of  the  first  order  in  the  whole  infinite  extent  of  the 
plane,  and  the  right  side,  how  often  this  function  becomes 
infinite  of  the  first  order;  from  this  follows  the  proposition: 
A  uniform  function  in  the  whole  infinite  extent  of  the  plane  is 
just  as  often  zero  as  it  is  infinite.  Whence  we  immediately 
infer :  A  uniform  function  assumes  every  arbitrary  value  k  just 
as  often  as  it  becomes  infinite. 

'FoT<t>(z)—k  becomes  infinite  as  often  as  <t)(z)  does;  hence 
<l>(z)—k  is  zero  just  as  often  as  <f)(z)  is  infinite,  and  therefore 
</>(2)  is  just  as  often  equal  to  k. 

From  this  follows  immediately  the  fundamental  proposition 
of  algebra;  for  an  integral  function  of  the  nth  degree  becomes 
infinite  only  for  z  =  oc,  and  that  of  multiplicity  n ;  therefore 


INF.  AND  INF'L    VALUES   OF  FUNCTIONS.         153 

it  must  also  become  7i  times  zero,  and  hence  an  equation  of  the 
nth  degree  must  have  n  roots. 

37.  We  can  now  prove  again  in  another  form  the  proposi- 
tion already  proved  in  §  32,  that  a  uniform  function,  which 
becomes  infinite  only  a  finite  number  of  times,  must  be  a 
rational  function. 

Let  tti,  a.2,  Og,  etc.,  be  the  finite  values  of  z  for  which 
a  uniform  function  <^(z)  becomes  infinite  or  infinitesimal, 
and  let  «i,  n2,  ng,  etc.,  respectively  denote  the  orders  of  the 

.  values,  positive  for  infinitesimal,  negative  for  infi- 

infinitesimal 

nite  values.     We  can  then  in  the  first  place  by  §  34  let 

^(z)  =  (2;-ai)"»tA(«), 

where   il/(z)   is  finite   and  not  zero  for  z  =  ai,  but  becomes 

infinite  ^^^  z  =  a^as,  etc.     Then 

infinitesimal 


where  now  ^i(^) 


,/,(2)  =  (z-a2)"^i(z), 
cf>(z) 


(z  —  ai)''i(z  —  ttj)"! 


does  not  become  .    „   .      .      ,  for  aj  and  a.,,  but  does  become 
infinitesimal 

^^     ^  ®  for   ttg,   etc. ;    if  we   continue   in  this  way,  we 

infinitesimal 

arrive  at  a  function 

^^^ ~(z-  a,yi{z  -  a,)"»(2  -  Og)"* . . .      U(z-ay 

which  no  longer  becomes   .    „   .      .      ,   for   any  finite  value 

infinitesimal 

of  z.     From  this  can  now  be  shown,  however,  that  it  cannot 

become  ^^^^^^^  .         for  z  =  oc.     For,  since 
infinitesimal 

(z-ar  =  z''(l-ff, 


we  can 


write  U(z  -  a)"  =  z^^ufl  -  -Y 


154  THEORY  OF  FUNCTIONS. 

But  if  m  denote  the  order  of  the  .    „   .      .      ,  vahie  of  4>(z) 

miinitesiinal 

for    z  =  oo,    positive    for    infinitesimal,   negative   for    infinite 

values,  then  by  (2),  §  36, 

Sft  =  —  m, 

since  here  2w  denotes  the  same  number  that  was  there  desig- 
nated by  2n  —  2v.     Accordingly  we  have 


and  X(z) 


u(z-ay=z-"'nfi--y 

z'^<j>(z) 


nfi-'* 


z 
But  for  2  =  00, 


lim  —    ^  \^^  =  lim z"'(f>(z), 


nfi-^ 


and  this  is  finite  and  not  zero  by  §  34,  since  d>(z)  is  .    ^   . 

infinitesimal 

of  the  mth  order  for  z  =  cc.      Therefore   X(z)   is   in  fact  a 

function  which  remains  finite  for  2  =  00;    now  since  it  also 

does  not  become  infinite  for  any  finite  value  of  z,  it  must  by 

§  28  be  a  constant.     If  we  denote  it  by  C,  we  have 

<f>(z)=  CU{z  -  ay. 

If  we  retain  now  aj,  ag,  Og,  etc.,  for  the  finite  values  of  z,  for 
which  cl}(z)  becomes  zero  of  orders  Ui,  no,  n^,  etc.,  respectively; 
and  if  we  denote  by  aj,  «2,  a^,  etc.,  the  finite  values,  for  which 
^(z)  becomes  infinite  of  orders  vi,  v^,  V3,  etc.,  respectively,  then 
we  have 

,    ^  (J  (g  -  ai)"<g  -  a^"^{z  -  a3)"3 .  •  ■  ^ 
^^  ^  (z-  aiY^iz  -  «2)-'2(2;  -  ag)''^  . .  . 

Therefore  ^(z)  is  actually  a  rational  function,  and  appears 
here  with  numerator  and  denominator  separated  into  factors, 
while  in  §  32  it  was  resolved  into  partial  fractions  and  an 
integral  function. 


INF.  AND  INF'L    VALUES  OF  FUNCTIONS.        155        J^  .  ^  cV 

<^^  ,0"^    <? 
From  this  follows,  further:  A  uniform  function  is  determine(Vy  ^     ' 
except  as  to  a  constant  factor,  when  once  we  know  all  finite  values  ^    /^* 
for  which  it  becomes  infinite  and  infinitesimal,  and  also  th^^  -S^ 

orders  of  the     -^  values;  and:   Two  uniform  functi^^, 

infinitesimal  Q* 

which  agree  in  these  values  and  in  their  orders,  are  equal  to  each 

other  except  as  to  a  constant  factor. 


B.  Functions  with  branchrpoints.     Algebraic  functions. 

38.  An  algebraic  function  w  of  2;  is  defined  by  an  algebraic 
equation,  in  which  the  coefficients  of  the  powers  of  w  are 
rational  functions  of  z ;  therefore  by  an  equation  of  the  form 

(1)  w^  -fAz)w^-'  +f{z)i<j^-- +(-  lX/;(z)=  0, 

wherein /i(2;), /2(z),  •••,  represent  rational  functions  of  z,  and  in 
which  the  coefficient  of  the  highest  power  of  w  is  assumed 
to  be  unity.  If  w^,  to^,  •••,  lo^  denote  the  p  roots  of  this  equa- 
tion for  any  assumed  value  of  z,  they  are  then  the  p  values  of 
the  function  for  the  value  of  z  in  question. 

Of  these  values  at  least  one  must  become  infinite  for  some  finite 
or  infinite  value  of  z.     For  we  have 

?(,j  4.  ji-^ -j t-^tv=/l(2). 

Now,  since /i(2;),  as  a  uniform  function,  must  by  §28  become 
infinite  for  some  value  of  z,  so  for  this  value  of  z  at  least  one 
of  the  summands  lOi,  w^,  •••,  ?t^  must  become  infinite. 

But  we  can  show  further  that  w  can  become  infinite  only  for 
such  a  value  of  z  as  leads  at  the  same  time  to  an  infinite  value 
of  at  least  one  of  the  rational  functions  fi(z),  f{z),  ••-.  For 
we  have 

Wi  +  ^t'2  -I h  iVp  =.fi(«) 

(2)  WiWi  +  wiWs  H h  «V-i«V  =  M^) 

WiW2-"Wj,=fp(z). 


156  THEORY  OF  FUNCTIONS. 

If  we  now  denote  by  w^  one  of  those  values  of  w  which  be- 
come infinite  for  a  certain  value  of  z,  we  can  remove  this  and 
introduce  the  sums  of  the  combinations  of  the  remaining 
values  of  w. 

Letting  Wi  +  Wj  +  •  •  •  +  ^^p-i  =  <^i(2!) 

wiW2  +  W1W3  -{ h  Wp^2iVj,_i  =  <t>2(z) 

we  then  easily  obtain 

fi{z)=<i>i(z)+Wp 

Mz)=  <i>2(z)-\-  lVj,<l>i(z) 


f„(z)  =  iVp<f>p_,{z). 

Now  since  Wp  is  infinite,  therefore,  by  reason  of  the  last  equa- 
tion, either  fp(z)  must  be  infinite,  and  then  the  above  proposition 
is  already  proved;  or,  if  this  be  not  the  case,  <j>p-i(z)  must 
vanish.  In  like  manner  it  follows  from  the  next  to  the  last 
equation,  that  either  fp.i{z)  must  be  infinite  or  ^^-2(2)  must  be 
zero.  Continuing  in  this  way  up  to  the  second  equation,  if  the 
case  occur  that  none  of  the  /-functions  from  fp(z)  to  /(z)  is 
infinite  for  the  value  of  z  in  question,  the  (^-functions  must  all 
vanish  from  <^p_i(z)  to  <^i(2),  and  therefore  it  follows  from  the 
first  equation  that  fi(z)  must  be  infinite. 

Now  since  the  converse  also  follows  from  equations  (2),  viz., 
that,  whenever  one  of  the  /-functions  becomes  infinite,  so  does 
at  least  one  of  the  w-functions,  we  obtain  all  the  z-points  at 
which  the  algebraic  function  w  becomes  infinite,  by  looking  for 
all  the  2;-points  at  which  the  rational  functions  /i(z), /(s;),  ••• 
become  infinite.  But  since  the  latter  can  become  infinite  at 
only  a  finite  number  of  points,  therefore  also  an  algebraic  func- 
tion can  become  infinite  at  only  a  finite  number  of  points} 

1  Konigsberger,  Vorlesungen  iiber  die  Theorie  der  elliptischen  Funk- 
tionen,  I.  S.  112. 


INF.  AND  INF'L    VALUES   OF  FUNCTIONS.         157 

It  can  now  be  proved  also  that  an  algebraic  function  cannot 
become  infinite  of  an  infinitely  high  order  at  any  point.  For, 
since  the  rational  /-functions  become  infinite  of  only  a  finite 
order  (§  32),  let  a  be  a  point  at  which  occurs  the  infinity  of 
highest  order  for  these  functions,  and  let  this  highest  order  be 
the  (r— l)th.  Then,  for  those  /-functions  which  become  infinite 
of  this  highest  order,  the  product 

is  neither  zero  nor  infinite  at  z  =  «  (§  29).  For  those  functions, 
on  the  other  hand,  which  are  either  not  infinite  at  all  or  infinite 
of  a  lower  order  at  z  =  u,  this  product  is  zero ;  and  this  .value 
holds  for  all  the  /functions,  when  in  place  of  ?•  —  1  a  higher 
exponent  occurs.  Now,  if  we  introduce  in  equation  (1)  another 
function  W  in  place  of  ?<;,  by  letting 

(3)  .  '^ 


we  obtain  for  TF"the  following  equation : 

W-iz-  ayfi{z)  W"-'  +(z-  aff^z)  W"-^ 

+  (-inz-arMz)=0. 

But  since  in  this  the  exponent  oi  z  —  a  for  every  coeflBcient 
is  greater  than  r  —  1,  these  coefficients  all  vanish  for  z  =  a,  and 

the  equation  reduces  to 

W''  =  0 

for  this  value  of  z,  so  that  the  values  of  W  corresponding  to 
z  =  a  are  all  zero.     Now,  from  (3)  follows 

W=(z  —  oCyw; 

therefore  the  values  of  the  w-functions  have  the  property  that 
for  them  the  product  (z  —  ayw  vanishes  at  the  point  z  =  a. 
But  since  one  or  more  of  them  are  here  infinite,  there  must 
be  for  them  a  positive,  integral  or  fractional  exponent  /i,  less 
than  r,  for  which  the  product  (z  —  «)'*w  receives  a  finite  value 


158  THEORY  OF  FUNCTIONS. 

different  from  zero;  and  in  such  a  case  we  again  say,  the 
wj-functions  involved  are  infinite  of  a  finite  order.^ 

If  a  denote  a  point  at  which  none  of  the  /-functions  acquires 
an  infinite  value  of  an  order  as  high  as  r  —  1,  but  at  which, 
nevertheless,  one  or  more  of  them  become  infinite,  the  state- 
ment holds  so  much  the  more.     But  if  an  infinite  value  occur 

for  2  =  00,  the  substitution  z  =  —  is  made,  and  then  the  /-func- 

u 

tions  become  rational  functions  of  u;  therefore  the  previous 
reasoning  is  applicable  to  the  point  w  =  0  if  the  w-functions  be 
also  treated  as  functions  of  u.  Hence  we  obtain  the  proposi- 
tion :  An  algebraic  function  always  becomes  infinite  at  a  finite 
number  of  points,  and  at  each  of  them  infinite  of  a  finite  order. 

In  the  Riemann  surface,  in  which  by  §  12  an  algebraic  func- 
tion can  be  regarded  as  a  uniform  function  of  position  in  the 
surface,  we  no  longer  need  to  examine  those  points  which  are 
not  branch-points,  since  for  them  the  principles  of  the  preced- 
ing section  which  refer  only  to  the  finite  parts  of  the  plane 
containing  no  branch-points  do  not  lose  their  validity. 

Hence  we  have  in  this  place  to  examine  in  detail  only  the 
branch-points  themselves,  and  we  begin  with  the  investigation 
made  in  §  21,  which  proved  the  following  proposition :  \i  z  =  b 
be  a  branch-point  of  a  function  /(z),  at  which  m  sheets  of  the 
z-surface  are  connected  [a  winding-point  of  the  (m  —  l)th  order 
(§  13)],  and  if  we  let  i 

{z-bf=l 

by  which  fiz)  changes  into  <^(^),  say,  then  <^(^)  does  noi  have 
a  branch-point  at  the  place  ^  =  0  corresponding  to  z  =  b. 

Now  we  can,  in  the  first  place,  apply  to  the  point  ^  =  0  the 
criterion  of  p.  128  for  the  finiteness  of  the  function,  and  infer 
that  </>  (^)  remains  finite  at  the  place  ^  =  0  if 

HmK</,(0]i^  =  0; 

therefore  we  obtain  as  the  necessary  and  sufficient  condition 
that  f(z)  remain  finite  at  the  branch-point  z=^b: 

lim[(2;-&)'^/(2)],,,  =  0. 
^  Konigsberger,  Vorlesungen,  u.  s.  w.,  I.  S.  177. 


INF.  AND  INF'L    VALUES  OF  FUNCTIONS.        159 

Further,  the  considerations  of  §  29  show  that  if  <f>  (^)  become 
infinite  of  the  «th  order  at  the  point  ^  =  0,  we  can  let 

a'     a"     a'"  a("> 

wherein  X  (t,)  remains  finite  for  ^  =  0,  and  the  quantities  g 
denote  constant  coefficients.     Therefore  we  have 

(z  —  6)"*      (z  —  6)"  (z  —  6)"* 

wherein  i/^  (z)  equals  X  (0,  say,  and  remains  finite  for  z  =  b. 
Then 

n 

lim  (z  —  b)'^f(z)  is  finite  and  not  zero,  and  the  order  oftlie  infinity 
of  f{z)  is  denoted  by  the  fraction  — 

At  b,  m  sheets  of  the  z-surface  are  connected,  hence  in  this 
place  m  function-values  become  equal.  If  these  be  designated 
\)j  w^Wi,  "•,  w„,  the  quantities 

w,(z  -  by,  lo^iz  -  by,  .'■,w„(z-  6)™ 

have  each  a  finite  limit  different  from  zero,  and  therefore  the 
same  is  also  true  of  the  product 

W1W2  ••'  w^(z  —  by. 

Hence  we  can  also  say:  The  function  w  becomes  infinite  of 
multiplicity  n  at  b  where  m  sheets  are  connected,  if  each  of  the 
values  becoming  equal  at  this  place  be  infinite  of  the  order  — 

The  principle  proved  on  p.  158  can  then  be  expressed  as  fol- 
lows :  An  algebraic  function  always  becomes  infinite  a  finite 
number  of  times. 

We  determine  more  explicitly  the  kind  of  infinity  of  f(z),  by 
specifying  the  expression  by  which  f(z)  differs  at  b  from  a 
function  which  remains  finite  at  that  point.     This  expression 


160  THEORY  OF  FUNCTIONS. 

proceeds,  as  the  last  equation  shows,  according  to  powers  of 

(2  — &)~™.     Thus,   we  say,   for  instance,    that  f(z)   becomes 

a'                      a"                      a'              a" 
infinite  as   — ^ — j-,  or  as  — ^ — ^,ot  as  — ^ — j-H -, 

g^g_  (z-6)'»  (2-6)^  (z-f.)-      (z-b)"' 

Let  us  now  consider  the  value  2  =  00,  which,  as  we  have 
already  seen,  §  14,  can  be  represented  by  a  definite  point,  and 
which  can  also  occur  as  a  branch-point.     Let  us  put 

2:  =  -  and  f(z)  =  (}i(u)  ; 
u 

then  w  =  0  is  a  branch-point  of  the  (m  —  l)th  order  for  <f)(u),  if 
z  =  00  be  such  ioTf(z).     Therefore /(z)  is  finite  for  2;  =  00,  if 


lim    u'^<j>{u)        =  lim 


\m 


=0. 


But,  if  f(z)  for  z  =  00 ,  and  hence  also  <j){u)  for  u  =  0  become 

infinite  of  the  order  — ,  then 
m 


Z"* 

is  finite  and  not  zero,  and  we  can  let 

a'       a"  a*-"^ 


or  /(2)  =  9'^  +  g"z^  +  ...-\-  g^"^z^  +  ^(2;),  (4) 

wherein  \j/(z)  =  A,  (u)  remains  finite  for  z  =  cc.     In  this  case  we 
say  that  /(z)  becomes  infinite  of  multiplicity  w  at  z  =  00 . 


39.  We  must  now  also  study  the  behavior  at  a  branch-point 

of  the  derivative  -^,  in  which  to  is  written  for/(z).     First  let 
dz 

us  consider  only  those  finite  points  at  which  w  remains  finite. 
It  has  been  proved  (§  24)  that  if  w  be  finite,  continuous  and 


INF.   AND  INF'L    VALUES  OF  FUNCTIONS.         161 

uniform  in  a  region,  i.e.,  if  it  possess  neither  points  of  discon- 
tinuity nor  branch-points,  then  —  likewise  remains  finite  and 

dz 

continuous  in  the  same  region.  Now,  if  we  express  the  deriva- 
tive by  the  limiting  value  to  which  it  is  equal,  denoting  by  w„ 
the  value  of  w  corresponding  to  z  =  a,  we  have 


g^=limh-^°1 
dz  |_  2  —  a  J, 


and  can  accordingly  say  that  when  z  =  a  is  neither  a  point  of 
discontinuity  nor  a  branch-point  of  w,  then 

lim 

is  not  infinite. 

But  we  can  also  determine  under  what  condition  this  limiting 
value  is  not  zero.  To  that  end  we  have  only  to  consider  z  as  a 
function  of  w.  If  the  point  w  =  w„,  corresponding  to  the  point 
z  =  a,  be  not  a  branch-point  of  the  function  z,  then  according 
to  the  above 


lim 


rz-al 
[jo  —  w„J, 


is  not  infinite,  and  hence  the  reciprocal  fraction 

limf^^^^ 

|_  z  —  a 

is  not  zero. 

Hence,  we  obtain  in  the  first  place  the  following  proposition : 
If  z  =  a  and  w  =  it\  be  two  finite  points  corresponding  to  each 
other,  and  if  neither  z  =  a  be  a  branch-point  of  w,  nor  w  =  ^t'„  a 
branch-point  of  z,  then 

lim  h-^"»1 

\_z-  a  J^ 

is  finite  and  not  zero. 

It  follows  that  the  derivative  —  at  a  finite  point  (at  which 

dz 

ic  also  is  finite)  can  become  zero  or  infinite,  only  when  at  that 


162 


THEORY  OF  FUNCTIONS. 


point  a  branching  occurs,  either  for  w  considered  as  a  function 
of  2,  or  for  z  as  a  function  of  w. 

Now,  if  in  place  of  a  a  branch-point  6  enter,  at  which,  how- 
ever, w  has  a  finite  value  Wj,  let 

the  z-surface  winding  m  times  round  b  (by  §  21) ;  then  w, 
considered  as  a  function  of  ^,  has  neither  a  point  of  discon- 
tinuity nor  a  branch-point  at  the  place  ^  =  0.  If  we  assume 
now  the  case  in  which  ^,  also  regarded  as  a  function  of  w,  does 
not  have  a  branch-point  at  the  place  w  =  w^,,  the  hypotheses 
of  the  preceding  proposition  are  satisfied,  and  therefore 


lim 


Wt 


=  lim 


i^ 


-  w  —  tVi,-i 
Jz  -  6)"!^ 


is  neither  zero  nor  infinite.     But  now 

(2-6)=^; 

therefore  z  is  a  rational  function  of  ^  and  consequently  by 
§  15  just  such  a  branched  function  of  w;  as  ^  is.  Therefore,  if 
^  do  not  possess  a  branch-point  at  the  place  w  =  w^,  then  z, 
considered  as  a  function  of  w,  likewise  has  no  such  point  there, 
and  hence  the  proposition  follows :  — 

(i.)  If  w  have  a  winding-point  of  the  (m  —  l)th  order  at  z=  b, 
hut  z,  regarded  as  a  function  of  w,  no  branch-point  at  tv  =  u\,  then 


lim 


^1 


is  finite  and  not  zero. 


\-(z  -  by 

If  this  finite  and  limiting  value  be  denoted  by  k,  then  also 


lim 


Rio  -  w^Y 

L  ^-^  - 


fc" 


but 


hence 


lim/"' 


w  —  ic. 


dz  \_(w  —  Wj)""^ 


dw 
dz 


=«j  L{z  —  6)~^-M» 


K 


INF.  AND  INF'L   VALUES  OF  FUNCTIONS.        163 

Therefore :  — 

(ii.)  With  the  hypothesis  of  proposition  (i.),  — ■  becomes  infinite 
at  b,  and  in  sttch  a  manner  thai 

liml  (w  —  w.)""*—  and  Um\  (2  —  6)  '»'— 

are  neither  zero  nor  infinite. 

T7,  s.r-    dw         1 

Ex.     io=^z,  — =  — _,  for  2  =  0. 
dz      3^ 

2 
If,  on  the  other  hand,  ^  or  {z  —  ft)"  possess  a  branch-point 

at  w  =  Wj,  such  that  p.  sheets  of  the  lo-surface  are  connected  in 

it,  the  hypotheses  of  proposition  (i.)  are  satisfied,  because  t,  as 

a  function  of  w  has  a  winding-point  of  the  (fi  —  l)th  order 

at  Wj,  but  w  as  a  function  of  ^  has  no  branch-point  at  ^  =  0 ; 

hence  lim > 

and  therefore  also  the  reciprocal  fraction 

iimR^-^\)n  , 

L    (z  —  Jm   J*^ 

is  finite  and  not  zero.     Since  now  z  and  ^  are  like-branched 
functions  of  w,  we  conclude :  — 

(iii.)  If  w  have  a  winding-point  of  the  (m  —  l)th  order  at 
z  =  b,  and  z  as  a  function  of  w,  a  winding-point  of  the  (jx  —  l)th 
order  at  the  corresponding  place  w  =  w^,  then 
I 

lim    ^ ^-       is  finite  and  not  zero. 

L  (z-  6)™  -l^» 

If  we  denote  this  finite  limit  by  h,  then 


^(w-w,y 
L  2-^ 


164  THEORY  OF  FUNCTIONS. 

and  since  lim =  — , 

|_  2  — 5  J^      dz 

also      —  =  lim  h'^(w  -  w^p^  =  lim    h>'{z  -  b)~^\     • 

Therefore :  — 

(iv.)  With  the  hypothesis  of  proposition  (iii.),  —  is  zero  or  infi- 

dz 

nite,  according  as  fi>or<m,  and  in  such  a  manner  that 
lim\  (w  —  wC)  '^  —  and  lim\  (z  —  b)  ""  — 

are  neither  zero  nor  infinite. 

Ex.     w^  =  2^,  for  2!  =  0;  m  =  S,  fji  =  2, 

dz         "7/;^  Z^ 

We  have  still  to  examine  the  value  z  =  x>,  retaining  the 
hypothesis  that  it  represents  a  branch-point  at  which  w  is 
finite.     Let 

1 

z  =  ~, 
u 

and  let  to'  be  the  value  of  w  corresponding  to  z  =  oo  or  w  =  0. 
If  we  assume  that  z  =  oo  is  a  winding-point  of  the  (m  —  l)th 
order  for  w,  but  that  w  =  w'  is  a  winding-point  of  the  (fi  —  l)th 
order  for  z,  we  obtain  by  (iii.),  since  z  and  u  are  like-branched 
fimctions  of  w,  and  also  since  the  branching  of  w  remains  the 
same :  — 

(y.)  Uml  (w  -^T        or  Um\  z^(w  -  wy 

is  finite  and  not  zero. 

K  this  limit  be  denoted  by  h,  we  have  by  (iv.) 


dio 
du 


=  lim   h'"{w  —  w^'^\       =  lim  Wu  "• 


INF.  AND  INF'L   VALUES  OF  FUNCTIONS.        165 

-D  J.  dw  1  dw 

But  now  —  = — ; 

dz  z^du 

therefore 


dw 
dz 


ft— nt  f* — n» 

Consequently :  — 

(vi.)  With  the  hypothesis  of  (v.),  —  is  zero,  and  in  sicch  a 
manner  that  the  expressions 

tf  tC^  dw  1  dw    J^  dw 

have  limits  finite  and  different  from  zero. 
Ex.     (w  —  w')'^  =  — ;  m  =  3,  /x  =  2; 

Finally,  let  us  turn  to  the  consideration  of  the  case  when  w 
itself  becomes  infinite  at  a  branch-point,  and  at  first  let  us 
assume  the  latter  to  be  finite  and  equal  to  b.  Now,  if  m  sheets 
are  connected  at  z  =  6  and  fx  sheets  at  w  =  oo,  we  can  deter- 
mine directly  from  (v.)  what  expression  remains  finite  and 
different  from  zero.  For,  if  z  —  6  be  there  put  in  place  of 
10  —  to',  and  lo  in  place  of  z,  and  if  further  m  and  fi  interchange, 
it  follows  that 


lim   w>^{z  —  6)"        =h 


remains  finite  and  not  zero.     Now  since  from  this  results 


lim  w{z  —  6)"        =  ^^ 


166  THEORY  OF  FUNCTIONS. 

SO  that  this  limit  is  also  neither  zero  nor  infinite,  it  follows 
(by  §  38)  that  w  is  infinite  of  the  order  —  in  this  case ;  and 
the  converse  at  the  same  time  holds.  Making  the  same  sub- 
stitutions as  above  in  the  second  of  the  expressions  (vi.),  we 

see  that 

1  dz 

(z  —  b)  ™ 


and  hence  also  the  reciprocal  value 

dz' 


is  finite  and  not  zero  in  the  limit,  and  that  therefore  —  is 
,  dz 

m  +  /u, 
infinite  of  the  order Hence  the  conclusion  is :     If  w 

become  infinite  of  the  order  —  at  a  winding-point  z  =  b  of  the 

(m  —  V)th  order,  then  the  point  tv  =  ao  itself  is  at  the  same  time 
a    branch-point    of  the   (fx  —  l)th  order,   and  conversely;   and 

-^r-  becomes  infinite  of  the  order -• 

dz  J  J  ^ 

Secondly,  if  w  become  infinite  for  z  =  <x> ,  and  if  this  point 

be  a  winding-point  of  the  (m  —  l)th  order,  while  w  =  oo  is  a 

winding-point  of  the  (yu,  —  l)th  order,  let  z  =  -;  then  by  the 
preceding  proposition 


and  hence 


lim  L 


».m 


-Z" 

is  finite  and  not  zero,  and  therefore  w  is  infinite  of  the  order 
-^.     Further, 


r  '^ 
hm    u 


liml.^^^^" 
du 


and   lim 


1    dw 

~^d^(. 


remain  finite  and  not  zero. 


INF.  AND  INF'L    VALUES  OF  FUNCTIONS.        167 

Now  since 

dw  2^w 

^ 2* 

du  dz' 


therefore  lim  |  ^  «.  _ 

dz 


.    r  ^:± 

lim      2;    m 


is  finite  and  different  from  zero,  and  hence  —  is  either  zero  or 

dz 
infinite,  according  as  m  >  or  <  fi.     E.g.,  take  the  equation 

(w  -  wy(z  -by  =  l; 


then 


w  —  w'  = ,  z  —  h  = 


{z-hy  (w-wy 

and  therefore  w  is  infinite  of  the  order  |,  for  z  =  b.  At  the 
same  time  three  sheets  of  the  ^-surface  are  connected  at  the 
place  z  =  b,  and  at  the  corresponding  place  w  =  00  five  sheets  of 
the  w-surface.     Further 

dw  __5  1 

^^      ^  (z~by 

and  therefore  the  derivative  is  infinite  of  the  order  |  for  z  =  b. 
For  the  equations 

w  =  z%    and  w  =  z^ , 

the  places  z  —  00  and  w  =  (xi  correspond.  We  obtain  respec- 
tively 

^  =  |1,   and  ^  =  y ; 

dz     5   i  dz      3     ' 

z 

hence,  for  2  =  00 ,  —  is  zero  in  the  first  case,  and  infinite  in 

dz 
the  second. 


40.  We  can  now  specify  in  what  way  the  2^surface  is  repre- 
sented on  the  w-surf  ace  in  the  vicinity  of  a  branch-point,  and 
thus  dispose  of  the  exceptional  case  mentioned  in  §  7. 


168  THEORY  OF  FUNCTIONS. 

If  we  assume  that  z  =  b  is  a  winding-point  of  the  (m  —  l)th 
order,  and  w  =  w^  a,  winding-point  of  the  (/u,  —  l)th  order,  we 
have,  by  (3),  §  39,  for 

(w  —  WjY 
1 
(z  —  6)"* 

a  definite,  finite  limiting  value  different  from  zero.  Therefore 
if  z'  and  z"  he  two  points  lying  infinitely  near  to  b  in  different 
directions,  w'  and  w"  the  points  of  the  w-surfaee  correspond- 
ing to  them,  then  it  is  the  above  expression,  and  no  longer,  as 

in  §  7,  the  expression  ^,  which  has  the  same  finite  limit 

z  —  h 

for  both  pairs  of  corresponding  points.     Therefore 

(w'—WjY       (w"—Wj,Y 

1  1 

(z'  -  ft)-       (z"  -  by 

i_ 

'z'-b 


or  fw'-w,y-^ 

\w"  —  wj        \z"  —  bj 

If  we  now  put  w'  —  Wj  =  p'  (cos  i/^'  +  %  sin  i/^'), 

w"  —  Wj  =  p"(cos  \p"  +  i  sin  i/^"), 

z'  —  b   =r'  (cos  <}>'  -t-  i  sin  <^'), 

z"-b    =r"(coscl>"  +  ism<f>"), 

we  have  ( -^  ]  (cos  ^  ~^   -f  %  sin  "^  ~  "^  ] 

\P  J  V  /*  I*-     / 


r'j  V  wi 


and  therefrom 

1 


-+i  sin ^—  b 

m    y 


p")   ~\y')  '        ^"~       m     ' 


or  iff^^iff'   ^("A'-"A")=)-(<^'-<^")- 


INF.  AND  INF'L    VALUES   OF  FUNCTIONS.         169 

Hence  there  no  longer  exists  similarity  in  the  infinitesimal 
elements  in  the  neighborhood  of  the  branch-point. 
In  the  example  cited  in  §  7, 


m  =  l  and  /x  =  2 ;  therefore  —  =  0,  for  the  branch-point  w  =  0 

(corresponding  to  z  =  0),  since  ft>m.     At  the  same  time  we 
have 

a  particular  case  (§  7). 

An  immediate  consequence  of  this  is  (among  others)  the 
proposition :  ^  TJie  angle  under  ichich  two  confocal  parabolas 
intersect  is  half  as  large  as  the  angle  between  their  axes.  By  the 
method  given  in  §  7,  or  also  easily  in  another  way,  we  satisfy 
ourselves  that  to  each  straight  line  in  z  which  does  not  pass 
through  the  origin,  corresponds  a  parabola  in  w,  the  focus  of 
which  is  at  the  origin,  and  the  axis  of  which  corresponds 
to  a  straight  line  in  z  passing  through  the  origin,  and  at  the 
same  time  parallel  to  the  former.  The  angle  which  two 
straight  lines  in  z,  not  passing  through  the  origin,  make  with 
each  other  is  just  as  large  as  the  angle  under  which  the  corre- 
sponding parabolas  intersect;  under  the  same  angle  also 
intersect  the  straight  lines  in  z,  passing  through  the  origin, 
which  correspond  to  the  axes  of  the  parabolas  in  w.  But 
since  the  origin  is  a  branch-point  of  z,  and  in  fact  m  =  1  and 
/i  =  2,  the  axes  of  the  parabolas  make  with  each  other  an 
angle  twice  as  large. 

41.  It  was  shown  above  (§  38)  that  a  multiform  algebraic 
function  becomes  infinite  a  finite  number  of  times.  We  now 
prove  the  converse,  namely : 

If  a  function  w  have  n  values  for  each  value  of  z,  and  become 
infinite  only  a  finite  number  of  times,  it  is  an  algebraic  function. 

1  Siebeck :  "  Ueber  die  graphische  Darstellung  imaginSrer  Fxmktionen," 
Crelle's  Jottrn.,  Bd.  55,  S.  239. 


170  THEORY  OF  FUNCTIONS, 

Let  us  denote  by  iCj,  iV2,  w^,  •••,  w„  tlie  n  values  of  w  corre- 
sponding to  a  definite  value  of  z.     If  we  form  the  product 

S  =(cr  —  Wi)(a  —  W2)  ••'  (o-  —  w„), 

wherein  a  denotes  an  arbitrary  quantity  independent  of  z, 
then  S  is  symmetric  with  regard  to  Wj,  W2,  •••,  w„.  Now  let 
z  describe  any  apparently  closed  line  (§  12),  then  some  or 
all  of  the  values  Wi,  Wg,  •••,  w„  will  have  changed,  but  at  the 
71  points  of  the  2!-surface  situated  one  immediately  above 
another,  w  will  again  have  the  same  values,  but  in  a  different 
sequence ;  consequently  S,  regarded  as  a  function  of  z,  has 
not  changed.  S  is  therefore  one-valued  at  all  points,  and 
hence  is  a  uniform  function  of  z.  In  addition,  S  becomes 
infinite  only  when  one  or  more  of  the  functions  Wi,  W2,  •••,io„  be- 
come infinite.  Each  of  the  latter,  according  to  the  assumption, 
becomes  infinite  only  a  finite  number  of  times ;  hence  the  same 
is  true  also  of  S.  Therefore  yiS  is  a  uniform  function  which 
becomes  infinite  only  a  finite  number  of  times ;  and  hence  by 
§  32  it  is  a  rational  function  of  z.  If  now  2  =  a  be  a  point 
of  discontinuity  of  \o  which  is  not  at  the  same  time  a  branch- 
point, and  if  w^  be  infinite  of  multiplicity  a  at  this  point,  then 

icjz  —  a)»,  and  therefore  also  (o-  —  w^{z  —  a)*, 

is  not  infinite  at  a  (§  29).  If,  further,  2;  =  6  be  at  the  same  time 
a  point  of  discontinuity  and  a  branch-point,  and  if  /x,  sheets  are 
connected  in  it,  then  p.  values  of  w  also  become  equal.  If  these 
be  denoted  by  Wi,  w^  •••,w;^,  and  if  the  number  of  times  that  w 
becomes  infinite  at  h  be  denoted  by  yS,  then  by  §  38  the 
quantities 

Wi(z  -  hy,  W2{z  -  by,  '■',  w^(z  -  by, 

and  therefore  also 

^  I  i. 

(or  -  wi)(z  -  by,  (a- -  W2)iz  -  by,--, (o- -  w^){z -  by, 

are  not  infinite.     Consequently  the  product 

(o- -  Wi)(o- -  W2)  •••  {a-w^){z-by 
also  remains  finite  for  z  =  b. 


INF.  AND  INF'L    VALUES  OF  FUNCTIONS. 


171 


Now  let 


0^1)  Otoj 


denote  the  points  of  discontinuity  which  are  not  branch-points, 
and 

the  points  of  discontintdty  which  are  at  the  same  time  branch- 
points ;  further,  let  the  respective  orders  a  and  (3  be  designated 


by  corresponding  subscripts.     Then 
product 


the  product 


X  (z  —  6i)^i(z  —  62)^2 

SZ  =  (o-  —  «-'i)(o-  —  tt'a) 
X  (z  —  ai)"i(2;  —  02)°* 
X  (2  —  61)^1  (z  —  62)^* 


if  we  multiply  S  by  the 

(2  -  a^yx. 
(z-6,)S 

(o-  —  w„) 
(z  -  a;,)-x 
(z  -  &,)^. 


remains  finite  for  all  values  a  and  6,  and  therefore  for  all 
finite  values  of  z.  Consequently  SZ  is  a  uniform  function 
which  becomes  infinite  only  for  z  =  00,  and  that  of  a  finite 
order ;  therefore  SZ  is  (by  §  31)  an  integral  function  of  z.  Now 
in  the  first  place  in  SZ  each  factor  of  Z  becomes  infinite  for 
z  =  00 ;  if  A  denote  the  number  of  times  that  w  becomes  infinite 
for  2  =  00,  then  the  number  of  times  that  SZ  becomes  infinite 
for  z  =  00  is 

and  this  number  is  exactly  the  number  of  times  that  w  becomes 
infinite  altogether.  For  lo  becomes  infinite  a  times  at  a  point 
a,  B  times  at  a  point  b,  and  h  times  at  the  point  z  =  00.     If  we  let 

/i  -i-  2«  +  2y8  =  m, 

then  SZ  is  an  integral  function  of  z  of  the  mth  degree.  Attend- 
ing now  to  the  quantity  o-,  we  see  that  SZ  is  also  an  integral 
function  of  o-  of  the  7tth  degree.  Therefore,  if  we  suppose  SZ 
to  be  arranged  according  to  powers  of  0-,  we  can  say  that  SZ  is 
an  integral  function  of  o-  of  the  nth  degree,  the  coefficients  of 


172  THEORY  OF  FUNCTIONS. 

which  are  integral  functions  of  z  at  most  of  the  mth  degree ; 
this  Riemann  was  in  the  habit  of  expressing  by  the  symbol 

Pi;,:)- 

This  expression  vanishes  when  <r  acquires  one  of  the  values 
Wi,  Wo,  •••,  w„,  and  hence  these  are  the  n  roots  of  the  equation 

Therefore :  An  n-valued  function,  which  becomes  infinite  of 
multiplicity  m,  is  the  root  of  an  algebraic  equation  between  w  and 
z,  of  the  nth  degree  with  regard  to  w,  the  coefficients  of  which  are 
integral  functions  ofz  at  most  of  the  mth  degree} 


SECTION  VIII. 

INTEGRALS. 


A.    Integrals  taken  along  closed  lines. 

42.  We  proceed  now  to  complete  the  propositions  given  in 
Section  IV.,  in  which,  however,  we  will  consider  only  infinite 
values  of  finite  order.  According  to  the  principles  relating  to 
infinite  values  of  functions  established  in  the  preceding  section, 
we  can  express  the  proposition  derived  in  §  20  in  the  form : 
If  the  integral 

]f(z)dz 


S^ 


be  taken  along  a  closed  line  enclosing  only  one  point  of  discon- 
tinuity a,  which  is  not  a  branch-point,  and  at  which  f(z)  becomes 
infinite  of  the  first  order,  then 


Cf(z)dz  =  2  TTt  lim  [(z  -  a)/(2)]. 


^  We  observe  that  here  the  coefficient  of  the  highest  power  of  w  is  not 
necessarily  unity,  as  was  assumed  in  §  38. 


INTEGRALS.  173 

We  will  now  investigate  the  value  of  this  integral,  when  f{z) 
is  infinite  of  the  nth  order  at  a.  By  §  29  we  have  in  the  domain 
of  the  point  a 

(1)  /(.)  =  ^-  +  ^-^  +  ...  +  -^  +  .-.+  -^  +  V'(^), 
z  —  a      {z  —  a)-  {z  —  ay  (z  —  a)" 

wherein  1/^(2)  remains  finite  and  continuous  for  z  =  a.     If  we 

now  construct   |  f(z)dz  in  reference  to  a  closed  line  round  the 

point  a,  we  can  choose  for  that  purpose  an  arbitrarily  small 
circle  described  round  a,  and  we  then  have  first 

Cil/(z)dz  =  0, 

and  in  addition  | =  2  iric'. 

J  z  —  a 

Next,  letting  z  —  a  =  r(cos  <i>  +  i  sin  <j>), 

.     r  c^^+^^dz      c(*+i't  r^',     , ,     .  .  ,  .. , , 

we  get       I = I     (cos  A;^  —  t  sin  k<f>)d(fi. 

J  {c  —  a)*+*        r*   c/o 

But  this  integral  vanishes,  because  for  every  integral  value  of 
k  not  zero 

J'^Zir  /»2jr 

cos  k<^d<^  =  0 ,    I     sin  kffsd^  =  0. 
0  Jo 

Therefore  for  every  value  of  k  different  from  unity 

J  (z-  a)" 

Therefore,  in  the  integration,  all  terms  except  the  first  vanish 
from  expression  (1)  and  we  have 


Cf(z)dz  =  2  Trie'. 


Accordingly  the  integral  is  always  equal  to  zero,  if  the  term 

c' 
be  wanting  in  the  expression  which  defines  the  nature  of 

z  —  a 

the  infinite  value  of  f(z);   but  if  this  term  be  present,  the 

integral  has  the  value  2  iric'. 


174  THEORY  OF  FUNCTIONS. 

Let  us  proceed  now  to  the  case  of  a  branch-point.  If  b  be 
a  point  of  discontinuity  at  which  m  sheets  of  the  z-surface  are 
connected,  we  have  in  the  vicinity  of  the  point  b  (§  38) 

a'  a"  o*"")  flW 

(2)     /(.)=_J^  +  -iL-,  +  ...+A_  +  ...+_9L_+... 

{z  -  6)"      (z  -  6)™  (z  —  by 

(z  -  &)•» 
wherein  ip(z)  is  finite  and  continuous  for  z  =  b.  If  we  now 
construct  i  f(z)dz,  taken  round  a  closed  line  enclosing  the 
point  b,  we  can  for  that  purpose  choose  an  arbitrarily  small 
circle,  the  circumference  of  which,  however,  must  be  described 
m  times  in  order  that  it  may  be  closed.  Again  in  the  first 
place 

Cil/(z)dz  =  0, 

and  further,  for 

z  —  b  =  ?*(cos  <jy-\-i  sin  <^), 

Cal^  =  C'"'g^'^md>  =  2  m7rta<»>. 
J  z  —  b     Jo 

Finally,  A;  denoting  an  integer  different  from  m, 

^  (z-  b)i  (z-  by^(z  -  b) 

_^w^  m    I       (  cos <^  —  I  sin d)  ]d<b. 

^  Jo      \  m  m        ) 

But  now  again 

cos <l>d^  =  0,     I       sm <f}d(f>  =  0, 

0  m  Jo  m 

as  long  as  k  is  not  equal  to  m,  and  hence,  also, 
•^  (z-6)» 


INTEGRALS.  175 

Therefore,  in  the  integration  of  expression  (2),  all  the  terms, 
with  the  exception  of  -^ — ,  vanish,  and  consequently 


Cf{z)dz  =  2  mnig^'^K 


Therefore,  this  integral,  likewise,  always  vanishes  when  the 

term  -^ is  wanting  in   the   expression  which   defines   the 

z  —  b 

nature  of  the  infinite  value  of  f{z),  and  the  proposition  in 
general  can  be  expressed  in  the  form :  — 

The  integral    i  f(z)dz,  taken  round  a  point  of  discontinuity 

about  which  the  z-surface  winds  m  times,  and  at  tvhich  f(z)  becomes 
infinite  of  a  finite  order,  has  a  value  different  from  zero,  when, 
and  only  when,  the  term  which  becomes  infinite  of  the  first  order 
is  present  in  the  expression  defining  the  nature  of  the  infinite 
value  off(z) ;  and  this  value  is  equal  to  2  mTri  times  the  coefficient 
of  this  term.  If  the  point  of  discontinuity  be  not  a  branch- 
point, we  have  only  to  let  m  =  1. 

43.  In  the  consideration  of  the  infinite  value  of  z,  we  have 
to  conceive  the  infinite  extent  of  the  plane,  by  §  14,  as  a  sphere 
with  an  infinite  radius,  therefore  as  a  closed  surface,  and  to 
imagine  the  value  z  =  cc  to  be  represented  by  a  definite  point. 
We  can  then  also  speak  of  closed  lines  which  enclose  the 
infinitely  distant  point.  We  will  now  investigate  the  behavior 
of  integrals  when  they  are  taken  round  such  closed  lines. 
These  still  form  closed  lines  when  we  imagine  the  infinite 
sphere  again  extended  in  the  plane,  but  then  that  region  which 
contains  the  point  z  =  cc  lies  in  the  plane  outside  the  line 
by  which  it  is  bounded. 

If  another  variable  u  be  introduced  instead  of  z,  by  letting 

z  —  h  = — 

u  —  li 

and  f{z)  =  <^(m), 

wherein  h  and  h  may  denote  two  points  to  be  chosen  arbitrarily, 
then  to  every  point  z  corresponds  a  point  u,  and  conversely. 


176  THEORY  OF  FUNCTIONS. 

But  to  the  points  z  =  h  and  u  =  k  correspond  respectively 

u  =  <xi  and  z  =  ao.     If  we  let 

z  —  h  =  r  (cos  <^  +  ^  sin  (ft), 

then  u  —  k  =  -  (cos  <f>  —  i  sin  <^). 

Now,  if  z  describe  a  closed  line  Z  enclosing  the  point  h,  then  <^ 
increases  from  0  to  a  multiple  of  2  tt  ;  hence  the  corresponding 
line  U,  described  by  u,  also  encloses  the  point  k,  and  indeed  in 
an  equal  number  of  circuits,  but  it  is  to  be  described  in  the 
opposite  direction.     Further,  if  z  go  from  the  perimeter  of  Z 

outward,  then  r,  or  modulus  oiz  —  h,  increases ;  therefore  -,  the 

r 

modidus  ot  u  —  k,  decreases,  and  hence  u  goes  from  the  perim- 
eter of  f7  inward.  Accordingly,  to  all  points  z  lying  without 
Z  correspond  such  points  u  as  lie  within  U.  If  we  now  regard 
the  curve  Z  as  the  boundary  of  the  portion  of  the  surface  lying 
on  the  outside,  the  positive  boundary-direction  for  this  is 
opposite  to  that  for  the  part  of  the  surface  in  the  interior; 
hence  Z  and  U  are  simultaneously  traversed  in  the  positive 
boundary-direction  of  corresponding  portions  of  the  surface. 

Now  f(z)  =  <l>(u),  dz  =  -      ^    A 

therefore  we  obtain 

wherein  the  first  integral  refers  to  the  curve  Z,  the  second  to 
the  corresponding  curve  U,  taken  over  both  in  the  positive 
boundary-direction.  Now,  if  there  be  in  a  closed  surface  a 
curve  Z  enclosing  the  point  oo,  this  becomes  in  the  plane  a 
closed  line  which  bounds  the  portion  of  the  surface  lying  on 
the  outside.  The  arbitrarily  assumed  point  h  can  always  be  so 
chosen  that  it  lies  within  the  curve  Z;  then  the  part  of  the 
surface  containing  the  point  z  =  ^  corresponds  to  the  part  of 
the  surface  lying  within  U,  and  the  above  equation 


Sma^^-m 


{n  -  kf 


INTEGRALS.  177 

holds,  extended  along  the  positive  boundaries  of  these  portions 
of  the  surface.      Thus   the  value  of  the   boundary   integral 

/f(z)dz  depends  upon  the  nature  of  the  function      *pW 

We  now  need  consider  only  such  curves  Z  as  contain  no  points 
of  discontinuity,  z  =  <x>  excepted ;  then  <^(m)  becomes  infinite 
within  U  at  most  for  u  =  k.     Thus  the  inquiry  comes  to  this, 

whether  and  how   ,  ^^  {     is  infinite  for  u  =  k.     This  expres- 

sion  is  equal  to  (z  —  hyf(z),  and  since  f or  2  =  oo 

\im{z  -  Jiffiz)  =  lim:^J{z), 

it  follows  that,  not  so  much  the  nature  of  the  function  f{z)  at  the 
point  z  =  cc,  as  much  more  that  of  the  function  z^f(z),  is  serviceable 
for  the  evaluation  of  the  boundary  integral.  But  if  this  principle 
be  observed,  all  previous  propositions  which  hold  for  boundary 
integrals  are  valid  also  for  such  closed  lines  as  enclose  the 
point  00 ;  at  the  same  time  it  is  to  be  kept  in  view,  however, 
that  when  the  integral  is  taken  in  the  positive  boundary- 
direction  of  the  piece  of  the  surface  containing  the  point  00 , 
the  value  of  the  integral  must  have  the  opposite  sign.  There- 
fore, if  z-f(z)  be  finite  for  z  =  00  ,  that  is,  if  lim  [;z/(^)]*^  =  ^> 
the  integral  is  zero ;  hence  it  does  not  sufiice  for  this  end  that 
f{z)  remain  finite,  the  function  must  rather  be  infinitesimal  of 
the  second  order.  Further,  if  z-f(z)  be  infinite  of  the  first 
order,  that  is,  if  lim  [z/(z)]^„  be  finite  and  not  zero,  then 


Jf(z)dz  =  -  2  TTJ  lim  [2/(2:)], 


the  integral  being  taken  in  the  positive  boundary-direction 
round  the  point  00  .  In  general,  the  integral  has  a  value  dif- 
ferent from  zero  when,  and  only  when,  in  the  development  of 
f(z),  in  ascending  and  descending  powers  of  z,  a  term  of  the 

form  ?  is  present. 


178  THEORY  OF  FUNCTIONS. 

dz 


Ex.  1. 


1+.^' 


here  lim  [2/(2)]^^  =  lim 


1+z' 


-0, 


therefore  the  integral,  taken  along  a  line  enclosing  the  point 
00 ,  is  equal  to  zero.  In  fact,  each  line  enclosing  the  two 
points  z  —  —  i  and  2  =  +  i  is  at  the  same  time  a  line  enclosing 
the  point  00  ,  since  the  function  has  no  other  points  of  discon- 
tinuity, and  we  have  already  seen  (§  20)  that  for  such  a  line 
the  integral  has  the  value  zero. 
Ex.  2.   If  the  integral 

''dz 


f 


be  taken  along  a  line  round  the  origin  in  the  direction  of 
increasing  angles,  it  has  the  value  2  vi.  But  the  same  line  is 
also   one  which   encloses   the   point  00 ,    since    the    function 

f(z)  =  -  possesses  only  the  one   point   of   discontinuity  z  =  0. 
z 

Now,  although  in  this  case  f(z)  is  not  infinite  for  z  =  cc ,  yet 
the  integral  has  a  value  different  from  zero,  because 


lim  [zf(z)],^  =  limfz  •  -  j 


We  therefore  obtain 

^dz 

z 


f" 


and  in  fact  the  line  must  be  described  in  the  opposite  direction 
if  it  bound  in  a  positive  direction  the  part  containing  the 
point  00 . 

Ex.  3.   We  can  from  this  principle  find  the  value  of 

^  Vl-z^ 

extended  along  a  line  running  in  the  first  sheet,  which  encloses 
the  two  discontinuity-  and  branch-points  + 1  and  —  1,  these 


INTEGRALS.  179 

being  joined  by  one  branch-cut.     For  such  a  linie  encloses  at 
the  same  time  the  point  oo ,  without  including  any  other  point 
of  discontinuity. 
In  this  example 


lim  [zf(z)],^  =  lim 
therefore 


Lvr 

J=±27r. 


where  the  sign  is  yet  to  be  determined.  But,  on  the  other 
hand,  the  line  which  encloses  the  points  +  1  and  —  1  can  be 
contracted  up  to  the  branch-cut.  If  we  then  agree  that  the 
radical  is  to  have  the  sign  -f  on  the  left  side  of  the  branch-cut 
(taken  in  the  direction  from  —  1  to  -f- 1)  in  the  first  sheet,  and 
hence  the  sign  —  on  the  right  side  of  the  same,  likewise  in  the 
first  sheet  (cf.  §  13),  then  also 


■=  f^'      f?^      _  r~'     dz       ^  2  f^ 

»/0 


dz 


vr^T^  j+i  vi^^    *^-^  vi  - 

dz 


integrated  in  the  direction  of  decreasing  angles  (in  the  posi- 
tive boundary-direction  round  the  point  oo).  Since  in  this 
case  all  the  elements  of  the  integral  are  positive,  J  must  also 
be  positive,  and  hence 

Jo  ^/i^^2 
and  therefore  also 

dz  V 


s 


With  respect  to  the  circumstance  that  the  integral  preserves  a 
finite  value,  although  the  function  — ^^=z  becomes  infinite  for 

z  =  1,  compare  the  following  paragraphs. 


180  THEORY  OF  FUNCTIONS. 


B.   Integrals  along  open  lines.     Indefinite  integrals. 

44.  We  will  inquire  in  this  paragraph  whether  and  under 
what  conditions  a  function  defined  by  an  integral  may  remain 
finite,  when  the  upper  limit  of  the  same  either  acquires  a 
value  for  which  the  function  under  the  integral  sign  becomes 
infinite,  or  the  limit  itself  tends  towards  infinity.  We  will 
inquire  further  in  what  manner  the  function  defined  by  the 
integral  becomes  infinite,  if  it  do  not  remain  finite  in  these 
cases.  But  at  the  same  time  we  limit  ourselves  to  such 
integrals  as  contain  algebraic  functions  under  the  integral 
sign. 

Let  F(t)  =  r<i>{z)dz 

be  the  integral  to  be  investigated,  wherein  h  denotes  an  arbi- 
trary constant.  We  here  consider  only  such  paths  of  integra- 
tion as  lead  to  the  same  value  of  the  function  ;  the  next  section 
will  show  that  the  multiformity  of  a  function  defined  by  an 
integral,  arising  from  different  paths  of  integration,  does  not 
affect  the  considerations  here  employed. 

If  we  assume  in  the  first  place  that  ^{z)  becomes  infinite  of 
the  nth  order  at  a  point  z  =  a,  which  is  not  a  branch-point,  we 
can,  by  §  29,  let 

(1)  ,^(,)  =  _^  +  _^  +  ...4-_2^  +  ^(^), 

z  —  a     {z  —ay  {z  —  ay 

wherein  \li(z)  remains  finite  for  z  =  a.     From  this  follows 

r<f>{z)dz = c'  f'-^ + c"  r--^+ ...  +c(")  c\  ^^ 

Jh  Jh  z  —  a         Jh  (z  —  ay  Jh  (z  —  ay* 

-\-£xp{z)dz. 

This  last  term  is  a  function  which  also  remains  finite  for 
f  =  a ;  if  we  denote  it  by  \(t),  and  if  we  suppose  included  in 


INTEGRALS.  181 

it  the  constant  terms  arising  from  the  lower  limits  h  of  the 
integrals,  we  then  obtain 

F(t)  =  c'log{t-a)- 


-a     2(t-ay  (n-l){t-ay~^ 

+  X{t). 

Now,  if  we  let  the  path  of  integration  end  in  the  point  t  =  a, 
then  the  function  defined  by  the  integral  is  distinguished  from 
a  function  k(t),  which  remains  finite  for  t  =  a,hy  a,  quantity 
which  contains  the  term  log  (t  —  a).  We  say  in  this  case,  F(t) 
becomes  logarithmically  infinite.     This  case  occurs  when  in  the 

c' 

expression  (1)  for  <\>{z)  the  term is  present.     If,  on  the 

z  —  a 

other  hand,  this  term  be  wanting,  the  logarithm  drops  out  and 
F{t)  becomes  infinite  of  an  integral  order.  But,  finally,  Fif) 
remains  finite  for  t  =  a,  only  when 

lim[(2;-a)«^(2)],^„=0; 

that  is,  when  <^(z)  itself  remains  finite  for  z  =  a. 

Next,  let  us  assmne  that  the  point  of  discontinuity  a  is  at 
the  same  time  a  branch-point.  If  m  sheets  of  the  2-surf ace 
be  connected  at  this  point,  we  can,  by  §  38,  let 

n'  o"  «(")  /j(™+J) 

(2)  *(.)  =  _iL^+^L-^+...  +  ^  +  _JL_^ 

(z  -  ay      (z  —  ay  '      (z  —  a)'^ 

wherein  \f/(z)  remains  finite  for  z  =  a.     From  this  we  obtain 

m  —  1  m  —  2 

1 
-I 1_  gr("-i)m(<  -  a)™  +  gr(»)  log  (t  -  a) 

(t  —  a)" 

if,  as  above,  X(t)  denote  the  last  term,  which  remains  finite, 
including  the  constants  arising  from  the  lower  limits  h. 


182  THEORY  OF  FUNCTIONS. 

If  at  most  the  first  m  —  1  terms  be  present  in  this  expression, 
F{t)  remains  finite  for  t  =  a.  This  case  occurs  when  in  (2) 
also  at  most  the  first  m  —  1  terms  are  present.    Then  ^(f)  is  at 

most  infinite  of  the  order ,  and,  therefore, 

m 

lim  [(2;  —  a)<^(2;)]^.„  =  0. 

Consequently  the  condition  that  F{t)  may  remain  finite  is  here 
the  same  as  before,  ^  and  the  general  propositions  follow :  — 

(i.)    Tlie  function  defined  by  the  integral 
F{t):=  C  <i>{z)dz 

of  an  algebraic  function  </>(z)  has  a  finite  value  for  t  =  a,  when, 

and  only  when, 

lim  [_(z  —  a)<f>(z)^^^^  =  0. 

(ii.)  If  lim  [(2  —  a)<ji(z)']^^a  ^^  finite  and  differeiit  from  zero, 
then  F(t)  is  logarithmically  infinite  for  t  =  a. 

(iii.)  If  lim  [(2  —  aY<i>{z)\^  have  a  finite  value  different  from 
zero  for  an  integral  or  fractional  exponent  p.,  which  is  greater 
than  unity,  then  F(t)  is  infinite  of  an  integral  or  fractional  order; 

and  if  in  the  development  of  ^(z)  the  term  of  the  form  — ^ —  be 

present,  F(t)  is  at  the  same  time  logarithmically  infinite. 

45.  We  have  now  to  examine  the  value  t  =  cc.  By  the  sub- 
stitution already  so  often  used 

1 

u 
we  reduce  this  case  to  the  former.     Let 

l  =  r,F(t)  =  F,(T),   <f>(z)=<l>,(u); 
then  ^(0  =/>(.)d.  =  -£^J^=F.ir). 

h 

1  We  note  in  particular :  If  the  function  </>  become  infinite  of  multi- 
plicity o  at  a  branch-point  of  the  (m  —  l)th  order,  and  a  <  m,  the  integral 
remains  finite. 


INTEGRALS. 


183 


The  nature  of  the  function  -F\(t)  depends,  therefore,  upon  the 

nature  of  the  fvmction       „  ^  for  the  value  u  =  0.     The  results 
w 

of  the  preceding  paragraph  then  give :  — 
(1)     -F\(t)  is  finite,  when 

lim 


=  lim 

u=0 

u 

=  0. 


(2)     Fx{t)  is  logarithmically  infinite,  when 


lim 


<^i(w)" 


is  finite  and  not  zero. 


(3)     Fi{t)  is  of  an  integral  or  fractional  order  (or  also  at  the 
same  time  logarithmically)  infinite,  when,  for  /x.>l, 


lim 


U'^<f)l(u) 


,     =lim  [tt^-2<^i(M)],^ 


is  finite  and  not  zero.  ^ 

Therefore  we  conclude,  for  t  =  cc:  — 

(i.)   F(t)  is  finite,  when  lim  [z<^(2:)]^^^=0. 

(ii.)   F(t)  is   logarithmically   infinite,   when   lim  [z(j>(z)^^^^  is 
finite  and  not  zero. 

(iii.)   F(t)  is  of  an  integral  or  fractional  order  (or  also  at  the 

same  time   logarithmically)  infinite  when  lim\  — ^         is  finite 

and  not  zero  (fi  positive  and  >1). 


Examples  : 
dz 


Jol 


+  z 
finite  for  t  —  cc 


is  logarithmically  infinite  for  t  =  ±i,  but  remains 


I    — — —  remains  finite  f or  i  =  ±  1,  and  is  logarithmically 

0      a/1    —22 


/o     -y/l—z 

infinite  for  t  =  cc 


184  THEORY  OF  FUNCTIONS. 


s: 


dz  1 

remains  finite  for  t  =  ±1  and.  t  =  ±  -, 


and  also  for  t  =  co;  hence  it  is  finite  for  every  value  of  t. 

Jr*  II  _  i^z^ 
■\ dz  remains  finite  for  ^  =  ±1,  and  becomes  in- 

finite  of  the  first  order  for  t=oc. 

I    =i=iz==^==  remains  finite  for  t  =  ±1, 

0    (1  -  a'z")  V(l  -  z^  (1  -  Jc'z^ 

and  t  =  ±-,  likewise  for  t  =  co,  and  becomes  logarithmically 

fC 

infinite  for  t  =  ±  -. 
a 


SECTION^  IX. 

SIMPLY    AND    MULTIPLY    CONNECTED    SURFACES. 

46.  For  the  investigation  of  the  multiformity  of  a  function 
defined  by  an  integral,  |  f(z)dz,  the  character  of  the  connec- 
tion of  the  z-surface,  for  the  function  f(z)  under  the  integral 
sign,  is  of  special  importance.  In  this  relation,  we  have 
already  pointed  out  (§  18)  the  marked  distinction  existing 
between  those  surfaces  in  which  every  closed  line  ^  forms  by 
itself  alone  the  complete  boundary  of  a  portion  of  the  surface, 
and  those  in  which  every  closed  line  does  not  possess  this 
property. 

We  call,  after  Riemann,  surfaces  of  the  first  kind  simply 
connected,  those  of  the  second  kind  multiply  connected.  A  cir- 
cular surface,  for  instance,  is  simply  connected;  so  is  the 
surface  of  an  ellipse,  and  in  general  every  surface  which  con- 
sists of   a  single  sheet   and   is   bounded  by  a  line  returning 

1  For  the  present  we  shall  understand  by  a  closed  line  such  a  one  as 
returns  simply  into  itself  without  crossing  itself. 


SIMPLY  AND  MULTIPLY  CONNECTED  SURFACES.     185 

simply  into  itself  without  crossing  itself.  Multiply  connected 
surfaces  can  arise  when  points  of  discontinuity  are  excluded 
from  simply  connected  surfaces  by  means  of  small  circles. 
For  instance,  if  we  exclude  from  a  circular  surface  a  point  of 
discontinuity  a,  by  enclosing  it  in  a 
small  circle  A;,  a  surface  is  formed 
which  is  no  longer  simply  con- 
nected ;  for,  if  a  closed  line  m  be 
drawn  round  k,  this  line  does  not 
constitute  the  complete  boundary 
of  a  portion  of  the  surface  by  it- 
self alone,  but  only  in  connection 
with  either  the  small  circle  k  or  the 
outer  circle  n.  But,  without  ex- 
cluding any  isolated  points,  we  may  have  multiply  connected 
surfaces,  when,  for  instance,  they  possess  branch-points,  and 
hence  consist  of  several  sheets  continuing  one  into  another 
over  the  branch-cuts. 

The  investigations  which  now  follow  relate  both  to  Rie- 
mann  surfaces  and  also  to  other  quite  arbitrarily  formed 
surfaces.  Nevertheless,  we  must  exclude  such  surfaces  as 
either  separate  along  a  line  into  several  sheets,  or  consist  of 
several  portions  connected  only  in  isolated  points  without 
winding  round  such  points,  as  the  Riemann  surfaces  do,  by 
means  of  branch-cuts.  For  surfaces  of  this  kind  (divided  sur- 
faces), the  properties  to  be  developed  would  not  be  valid  in 
their  full  extent.  But  since  we  have  here,  nevertheless,  to  do 
with  surfaces  the  structures  of  which  can  be  extraordinarily 
manifold,  we  must  seek  to  base  our  investigations  as  much  as 
possible  upon  general  considerations. 

In  the  first  place,  it  is  important  to  obtain  a  definite  criterion 
by  which  we  can  distinguish  whether  or  not  a  closed  line 
forms  by  itself  alone  the  complete  boundary  of  a  portion  of 
the  surface.  To  this  end,  we  remark  that  two  portions  of  a 
surface  are  said  to  be  connected  when,  from  any  point  of  one 
portion  to  any  point  of  the  other,  we  can  pass  along  a  contin- 
uous line  without  crossing  a  boundary-line;   in  the  opposite 


186  THEORY  OF  FUNCTIONS. 

case,  the  portions  of  the  surface  are  said  to  be  distinct.  If  a 
portion  A  of  the  surface  be  completely  bounded,  it  must  be 
separated  by  its  boundary  from  the  other  portion  B  of  the 
surface ;  otherwise,  we  could  pass  from  ^  to  jB  without  cross- 
ing the  boundary  of  A,  and  hence  that  boundary  would  not  be 
complete. 

We  will  assume  that  the  surface  to  be  considered  is  bounded 
by  one  or  more  lines,  that  it  possesses  an  edge  consisting  of 
one  or  more  boundary-edges.  We  will  always  assume  that 
these  lines  return  simply  into  themselves  and  nowhere  branch.  In 
the  Riemann  surfaces,  this  is  always  the  case,  since  in  these 
a  boundary-line  has  only  one  definite  continuation  at  every 
place,  even  where  it  passes  into  another  sheet ;  but  in  divided 
surfaces  this  would  not  always  hold.  In  order  that,  within 
such  a  surface,  a  closed  line  m  may  form  by  itself  alone  the 
complete  boundary  of  a  portion  of  the  surface,  the  following 
condition  is  necessary  and  sufficient.  By  the  line  m  a 
piece,  containing  none  of  the  original  boundary-lines,  must  be 
separated  from  the  given  surface.  We  can  now  show  that  this 
condition  is  satisfied,  when  we  can  come  from  any  point  of 
the  line  m  to  the  edge  of  the  surface,  without  crossing  the  line 
m,  only  on  one  side ;  that,  on  the  contrary,  when  this  is  possi- 
ble on  each  side  of  the  line  m,  the  latter  cannot  form  a  com- 
plete boundary. 

For,  if  we  suppose  the  surface  actually  cut  along  the  line  m, 
two  cases  are  possible:  either  the  surface  is  divided  by  the 
section  into  distinct  pieces,  or  it  is  not.  In  the  latter  case,  no 
part  is  separated  from  the  surface,  and  therefore  m  cannot  form 
the  boundary  of  a  piece.  Since,  however,  in  this  case  all  por- 
tions of  the  surface  are  still  connected,  we  can  come  from 
either  side  of  m  to  the  boundary  of  the  surface. 

If,  in  the  opposite  case,  the  surface  be  divided  by  the  section 
along  m  into  distinct  pieces,  it  reduces  to  only  two  pieces,  A 
and  B,  because  an  interruption  of  the  connection  has  nowhere 
occurred  along  one  and  the  same  side  of  m.  Now,  either 
both  pieces  A  and  B  can  contain  original  boundary-lines,  or 
only  one  of  these  pieces  can.     If  both  contain  boundary-lines. 


SIMPLY  AND  MULTIPLY  CONNECTED  SURFACES.     187 

neither  of  them  is  bounded  by  m  alone ;  in  this  case,  we  can 
again  come  from  m  to  an  edge  of  the  surface  on  each  side  of 
m.  If,  on  the  other  hand,  only  the  one  piece  B  contain  one  or 
more  boundary-lines,  and  the  other  piece  A  not,  then  m  forms 
by  itself  alone  the  complete  boundary  of  A,  and  we  can  come  to 
the  edge  of  the  surface  only  on  one  side  of  m,  namely,  in  B, 
not  on  the  other  side  in  A.  Consequently  a  closed  line  m 
does,  or  does  not,  actually  form  a  complete  boundary  by  itself 
alone,  according  as  we  can  come  from  vi  to  a  boundary  of  the 
surface  only  on  one  side,  or  on  each  side  of  m.  (Cf.  Fig.  36, 
where  we  can  come  from  an  arbitrary  point  of  the  line  m  on 
the  one  side  to  the  part  of  the  boundary  k,  on  the  other  side  to 
the  part  of  the  boundary  n.) 

This  criterion  cannot  be  immediately  applied  to  completely 
closed  surfaces,  which,  as  for  instance  a  spherical  surface,  do 
not  possess  a  boundary.  But  we  can  assign  a  boundary  to 
such  a  surface  by  taking  at  any  place  an  infinitely  small  circle, 
or  what  is  the  same,  a  single  point  as  boundary.  (We  suppose 
a  sheet  of  the  surface  pricked  through  with  a  needle  at  some 
point.)  This  point,  or  the  circumference  of  the  infinitely 
small  circle,  then  constitutes  the  boundary  or  edge  of  the 
surface.  We  shall  always,  hereafter,  suppose  a  closed  sur- 
face bounded  in  this  way  by  a  point,  which  can,  moreover, 
be  assumed  in  any  arbitrary  place  in  the  surface.  By  this 
means  the  above  criterion  also  becomes  applicable  to  closed 
surfaces.  We  may  now  adduce  some  examples  in  illustration 
of  the  preceding. 

(i.)  A  spherical  surface  is  simply  connected.  For,  if  we 
draw  in  it  any  closed  line  m,  and  assume  anywhere  in  the 
surface  a  point  x  as  boundary,  we  can  always  come  to  x  from 
m  only  on  the  one  side,  never  at  the  same  time  on  the  other 
side ;  therefore  every  closed  line  m  forms  by  itself  alone  a 
complete  boundary. 

(ii.)  If  a  surface  have  a  branch-point  a,  at  which  n  sheets 
of  the  surface  are  connected,  and  if  a  portion  of  the  surface  be 
bounded  by  a  line  making  n  circuits  round  the  point  a,  and 
therefore  being  closed,  the  portion  of  the  surface  so  bounded  is 


188 


TUEOBY  OF  FUNCTIONS. 


simply  connected.  For  in  whatever  way  we  may  draw  therein 
a  closed  line,  we  can  always  come  only  on  the  one  side  of  the 
same  to  the  edge  of  the  surface. 

(iii.)   A  surface  consisting  of  two  sheets  closed  at  infinity, 
and  possessing  two  branch-points  a  and  b  (Fig.  37),  which  are 

connected  by  a  branch-cut, 
is  a  simply  connected  sur- 
face. We  can  in  this  case 
draw  only  three  different 
kinds  of  closed  lines  :  such 
as  enclose  no  branch-point, 
such  as  enclose  one,  and 
such  as  enclose  two.  The 
first  and  last  kinds  are  not 
essentially  different  from 
each  other;  for,  according 
as  we  regard  such  a  line  m 
or  n  as  the  boundary  of  the 
inner  or  outer  portion  of  the  surface,  it  encloses  either  both 
branch-points  or  neither.  But  such  a  line  as  m  or  n  always 
forms  a  complete  boundary,  for  we  can  always  come  from  it  to 
the  arbitrarily  assumed  boundary-point  only  on  the  one  side. 
A  finite  closed  line  enclosing  only  one  branch-point,  for  instance 
a,  goes  twice  round  the  same,  because 
in  crossing  the  branch-cut  it  enters  the 
second  sheet,  and  therefore,  in  order  to 
return  into  the  first  and  become  closed, 
it  must  again  cross  the  branch-cut.  But 
then  it  likewise  forms  a  complete  bound- 
ary. 

(iv.)   The  preceding  surface  becomes 

multiply   connected,   when    once    it   is 

bounded  in  each  sheet  by  a  closed  line 

(h  and  k,  Fig.  38 ;  the  dotted  line  runs 

in  the  second  sheet) ;  for  now  a  line  enclosing  a  and  b  in  the 

first  sheet  does  not  form  a  complete  boundary,  because  we  can 

come  from  it  to  the  edge  of  the  surface  on  each  side ;  namely. 


SIMPLY  AND  MULTIPLY  CONNECTED   SURFACES.     189 

on  the  one  side   directly  to  k,  on  the   other   to   h  over  the 
branch-cut. 

(v.)  A  surface  consisting  of  two  sheets,  closed  at  infinity,  and 
possessing  four  branch-points  joined  in  pairs  by  brauch-cuts  ab, 
cd,  is  multiply  connected  (Fig.  39).  For,  if  we  draw  a  line  m, 
enclosing  the  points  a  and  b  in  the  first  sheet,  we  can  come  from 
the  same  to  the  arbitrarily  assumed  boundary-point  x  on  each 
side.  If  X  be  in  the  first  sheet,  say,  this  is  done  directly  on  the 
one  side,  on  the  other, 
however,  by  crossing 
the  branch-cut  ab.  By 
this  means  we  arrive 
in  the  second  sheet  and 
can,  without  meeting 
the  line  m  (since  this 
runs  entirely  in  the  first 
sheet),  come  to  the 
other    branch-cut    cd ,  ^^'^-  ^^• 

crossing  it,  we  return  again  into  the  first  sheet  and  so  arrive, 
as  before,  at  x. 

47.  Now  it  is  of  the  greatest  importance  that  we  be  able 
to  modify  a  multiply  connected  surface  into  one  simply  con- 
nected by  adding  certain  boundary-lines.  As  will  appear  later 
(§  56),  this  is  always  possible  in  a  Riemann  surface,  if  it  have 
a  finite  number  of  sheets  and  branch-points,  and  if  its  boundary- 
lines  form  a  finite  line-system  (in  the  meaning  of  §  50).  These 
new  boundary-lines  are  called,  after  Eiemann,  cross-cuts.  That 
is,  by  a  cross-cut  is  understood  in  general  a  line  which  begins 
at  one  point  of  a  boundary,  goes  into  the  interior  of  the  surface 
and,  without  anywhere  intersecting  either  another  boundary- 
line  or  itself,  ends  at  a  point  of  the  boundary.  In  order  that 
the  meaning  and  extent  of  this  definition  may  be  made  perfectly 
clear,  let  us  consider  somewhat  more  in  detail  the  different 
kinds  of  cross-cuts.  A  cross-cut  can  connect  with  each  other 
two  points  of  the  same  boundary-line  {ab,  Fig.  40) ;  also,  two 
points  (cd)  situated  on  different  boundary-lines.     It  can  also 


190  THEORY  OF  FUNCTIONS. 

end  in  the  same  point  of  a  boundary-line  in  which  it  began 
{efge),  and  therefore  be  a  closed  line.  This  is  especially  the 
case,  when  a  cross-cut  is  to  be  drawn  in  a  closed  surface ;  for, 
since  in  such  a  surface  the  original  boundary  consists  of  only 
a  single  point  (§  46),  the  cross-cut  must  begin  in  this  point  and 
also  end  in  it,  unless  the  case  to  be  immediately  mentioned 
occurs,  in  which  it  ends  in  a  point  of  its  previous  course.  It 
has  been  stated  already  that  the  cross-cuts  are  to  be  regarded 
as  boundary-lines,  added  to  the  already  existing  boundary -lines. 
Hence,  if  a  cross-cut  have  been  begun,  its  points  are  immediately 
regarded  as  belonging  to  a  newly  added  boundary ;  and  since 
it  is  only  necessary  for  a  cross-cut  to  end  in  a  point  of  a  bound- 
ary, it  can  also  end  in  one  of  its  previous  points  (abed,  Fig.  41). 


Fig.  41. 


For  the  same  reason,  since  each  cross-cut  already  drawn  forms 
part  of  the  boundary,  a  subsequent  cross-cut  can  begin  or  end 
at  a  point  of  a  previous  one.  (Fig.  41,  where  ef  is  a  pre- 
vious cross-cut,  and  gh  a  subsequent  one.)  Finally,  stress  is  to 
be  laid  upon  the  following  consideration.  Since  a  cross-cut  is 
never  to  cross  a  boundary-line,  it  is  also  never  to  cross  a 
previous  cross-cut.  Therefore,  if  a  line  joining  two  boundary- 
points  cross  a  previous  cross-cut,  such  a  line  forms  not  one, 
but  two  cross-cuts ;  since  one  ends  at  the  point  of  intersection, 
and  at  the  same  point  a  new  one  begins.  Thus,  for  instance, 
in  Fig.  42,  the  two  lines  ab  and  cd  form  not  two,  but  three 


SIMPLY  AND  MULTIPLY  CONNECTED   SURFACES.     191 


cross-cuts;  namely,  according  as  ah  or  cd  was  first  drawn, 
either  ah,  ce,  ed,  or  cd,  ae,  eh.  In  like  manner,  two  cross-cuts 
are  formed  by  the  line  fghi,  namely,  fghg  and  gi,  or  ighg  and  gf. 

In  all  cases  a  cross-cut  is  to  be  regarded 
as  a  section  actually  made  in  the  surface, 
so  that  in  it  two  boundary-lines  (the  two 
edges  produced  by  the  section)  are  always 
united,  one  of  which  belongs  as  bound- 
ary-line to  the  portion  of  the  surface 
lying  on  one  side  of  the  cross-cut,  the 
other  to  that  on  the  other  side. 

The  possibility  of  modifying  multiply 
connected  surfaces  into  simply  connected 
surfaces  may  be  brought  into  consideration 
in  the  first  place  in  some  simple  cases. 
For  instance,  if  a  cross-cut  be  be  drawn  in 
the  surface  bounded  by  the  lines  k  and  n 
(Fig.  43),  and  if  both  sides  of  the  same  be  included  in  the 
boundary  (since  the  surface  is  regarded  as  actually  cut  through 
along  6c),  a  closed  line  can  no   longer  be  drawn  to  include 


Fig.  42. 


■n  t 


3^ 


Fig.  48. 


Fig.  44. 


k,  but  every  closed  line  forms  by  itself  alone  a  complete 
boundary.  The  same  condition  is  obtained  in  the  surface 
bounded  by  the  lines  h,  k,  n  (Fig,  44)  by  means  of  the  cross- 
cuts ah,  cd.  We  remark  that,  in  the  last  example,  the  modi- 
fication into  a  simply  connected   surface  can  be  effected  in 


192 


THEORY  OF  FUNCTIONS. 


several  ways,  but  always  by  means  of  two  cross-cuts  ab,  cd; 
for  example,  as  in  Fig.  45  and  Fig.  46. 


Fio.  40.  Fig.  46. 

48.  We  now  proceed  to  the  general  investigation  of  the 
resolvability  of  a  multiply  connected  surface  into  one  simply 
connected,  and  to  that  end  we  prove  some  preliminary  proposi- 
tions. 

I.  If  a  surface  T  be  not  resolved  by  any  cross-cut  ab  into  dis- 
tinct pieces,  it  is  multiply  connected. 

Let  us  first  assume  that  the  end-points  a  and  b  of  the  cross- 
cut both  lie  on  the  original  boundary  of  T;  by  this  assumption, 
however,  we  are  not  to  exclude  the  case  in  which  a  and  b 
coincide.  Since,  according  to  the  hypothesis,  the  cross-cut  ab 
does  not  divide  the  surface  into  distinct  pieces,  the  two  sides  of 
the  same  are  connected,  and  a  closed  line  m  can  be  drawn  from 
a  point  c  on  the  cross-cut  which  leads  from  one  side  of  it  through 
the  interior  of  the  siirface  to  the  other  side.^  Such  a  closed  line 
m,  however,  does  not  form  by  itself  alone  a  complete  boundary ; 
for  we  can  come  from  c  on  each  side  of  m  along  the  cross-cut 
to  the  edge  of  T,  that  is,  to  a  and  b.  Therefore  T  is  in  fact 
multiply  connected.  The  same  is  true,  if  the  cross-cut,  not 
resolving  the  surface,  be  such  a  one  as  ends  in  a  point  of  its 
previous  course  (cf.  Fig.  41,  abed).  For,  in  that  case,  we  must 
be  able  to  draw  from  a  point  c,  situated  on  the  closed  part  of 
the  cross-cut,  a  closed  line  m  leading  from  the  one  side  of  the 


^This  construction  is  to  be  so  understood  here,  and  likewise  also  later, 
that  the  line  m  would  be  closed,  if  the  cross-cut  did  not  exist. 


SIMPLY  AND  MULTIPLY  CONNECTED   SURFACES.     193 

same  to  the  other  side.^  But  such  a  closed  line  m  does  not 
form  a  complete  bovmdary,  since  we  can  come  from  c  on  each 
side  of  m  along  the  cross-cut  to  the  edge  of  the  surface,  that 
is,  to  a.     (In  Fig.  41,  the  two  paths  are  cba  and  cdba.) 

II.  It  is  alivays  jwssible  to  draw,  in  a  multiply  connected  sur- 
face, at  least  one  cross-cut  which  does  not  resolve  the  surface  into 
distinct  ^yieces. 

Since  the  surface  is  multiply  connected,  there  is  in  it  at  least 
one  closed  line  m  which  does  not  form  by  itself  alone  a  com- 
plete boundary ;  thus  we  can  come  from  each  side  of  this  lines  to 
the  edge  of  the  surface  (§  46).  "VYe  can  therefore  draw  from  a 
point  c  of  the  line  m  two  lines,  ca  and  cb,  which  go  on  different 
sides  of  the  line  m  through  the  interior  of  the  surface  and  end 
in  the  points  a  and  b  of  the  edge  (wherein  a  and  b  may  also 
coincide).  These  two  lines  then  form  together  a  cross-cut  ab, 
because  together  they  may  be  regarded  as  one  line  which 
begins  in  a  point  a  of  the  edge  and,  without  anywhere  crossing 
a  boundary  line,  ends  in  a  point  b  of  the  edge.  This  cross-cut 
does  not,  however,  resolve  the  surface,  for  we  can  come  along 
the  line  m  itself  from  the  one  side  of  the  cross-cut  to  the  other 
side  of  the  same ;  so  that  these  two  pieces  of  the  surface  are 
connected  and  not  distinct. 

Note.  — The  foregoing  shows  at  the  same  time  how  we  can  draw  in  a 
multiply  connected  surface  a  cross-cut  which  does  not  divide  the  surface, 
when  we  know  in  the  surface  a  closed  line  which  forms  by  itself  alone  a 
complete  boundary. 

III.  A  surface  consisting  of  one  piece  can  be  resolved  at  most 
into  two  pieces  by  a  cross-cut. 

Either  the  portions  of  the  surface  lying  on  each  side  of  the 
cross-cut  are  connected,  in  which  case  the  cross-cut  does  not 
resolve  the  surface ;  or  they  are  not  connected,  in  which  case 
those  portions  of  the  surface  lie  in  distinct  pieces.  If  the 
number  of  the  latter  amount  to  more  than  two,  there  must 

iThis  is  possible  when,  for  instance,  within  the  closed  portion  bed,  there 
is  a  branch-cut  which  the  line  m  can  cross,  thereby  coming  into  another 
sheet  in  which  it  does  not  meet  the  cross-cut,  and  when  by  means  of  a 
second  branch-cut  it  can  return  to  the  initial  point. 


194  THEORY  OF  FUIfCTIONS. 

occur  an  interruption  of  the  connection  in  a  portion  of  the 
surface  lying  on  one  and  the  same  side  of  the  cross-cut ;  but 
this  is  not  the  case,  because  the  cross-cut  nowhere  crosses  a 
boundary-line.^ 

IV.  A  simply  connected  surface  is  resolved  by  every  cross-cut 
into  two  distinct  pieces,  each  of  wJiich  is  again  by  itself  simply 
connected. 

The  first  part  of  the  proposition  follows  immediately  from 
I.  and  III. ;  for,  if  no  cross-cut  would  resolve  the  surface, 
the  latter  could  not  be  simply  connected,  and  it  cannot  be 
resolved  into  more  than  two  pieces.  But  it  is  evident  with- 
out further  proof  that  each  of  the  pieces  formed  must  be 
by  itself  simply  connected ;  for,  since  in  the  unresolved 
surface  every  closed  line  forms  by  itself  a  complete  boundary, 
the  same  also  holds  of  every  closed  line  which  runs  entirely 
within  one  of  the  pieces  formed. 

V.  A  simply  connected  surface  is  resolved  by  q  cross-cuts  into 
q-\-l  distinct  pieces,  each  of  which  is  by  itself  simply  connected. 

If,  after  the  surface  is  first  resolved  into  two  distinct  pieces 
by  one  cross-cut  (IV.),  a  new  cross-cut  be  drawn,  this  can  run 
in  only  one  of  the  two  pieces  already  formed,  because  it  is  not 
allowed  to  cross  the  first  cross-cut  (§  47) ;  but  it  resolves  the 
portion  in  which  it  falls  into  two  pieces,  so  that  the  two  cross- 
cuts divide  the  surface  into  three  pieces.  These  again  are  each 
by  itself  simply  connected.  If  we  now  draw  a  new  cross-cut, 
again  only  one  of  the  already  existing  pieces  is  resolved; 
and  likewise  the  number  of  pieces  is  increased  only  by  unity 
for  every  succeeding  cross-cut.  Therefore,  at  the  end,  after  q 
cross-cuts  have  been  drawn,  q  -{-1  distinct  pieces  are  formed ; 
and  these  are  each  by  itself  simply  connected  (IV.). 

Cor.  From  this  follows  immediately :  If  there  be  a  system 
of  surfaces  consisting  of  a  distinct  pieces,  each  by  itself  simply 
connected,  this  is  resolved  by  q  cross-cuts  into  a-\-  q  simply 
connected  pieces. 

1  In  the  case  of  a  cross-cut  ending  in  a  point  of  its  previous  course,  the 
one  side  consists  of  the  portions  of  the  surface  on  the  inside  contiguous  to 
the  closed  part,  the  other  side  of  the  remaining  portions  of  the  surface. 


SIMPLY  AND  MULTIPLY  CONNECTED  SURFACES.     195 

Note.  —  The  foregoing  considerations  are  also  applicable  to  the  case  in 
which  the  a  pieces  of  which  the  original  system  consists  are  not  all  simply 
connected.  In  that  case,  however,  the  distinction  occurs,  that  there  may 
now  be  cross-cuts  not  resolving  the  pieces  in  which  they  are  constructed  ; 
and  finally,  as  a  consequence,  that,  after  the  introduction  of  the  q  cross- 
cuts, less  than  a+q  distinct  pieces  are  formed.  But  more  than  a  +  q  pieces 
cannot  arise.  We  conclude  thei-efore  :  If  q  cross-cuts  be  drawn  in  a 
system  of  surfaces  consisting  of  a  pieces,  the  number  of  pieces  arising 
thereby  is  either  equal  to  or  less  than  a  +  q,  but  never  greater  than 
a  +  q. 

VI.  If  a  surface  he  resolved  by  every  cross-cut  into  distinct 
pieces,  it  is  simply  connected. 

For,  were  it  multiply  connected,  it  could  not  be  resolved  by 
every  cross-cut  (II.). 

VII.  //'  a  surface  T  he  resolved  hy  any  one  definite  cross-cut  Q 
into  two  distinct  pieces,  A  and  B,  each  of  which  is  hy  itself  simply 
connected,  T  is  also  simply  connected. 

We  shall  show  that,  under  the  given  hypothesis,  every  cross- 
cut drawn  in  T  must  resolve  this  surface.  In  the  first  place, 
it  is  evident  that  every  cross-cut  which  lies  entirely  within  A 
or  B,  and  which  therefore  does  not  cross  Q,  resolves  the  sur- 
face ;  for,  if  such  a  cross-cut  lie  entirely  within  A,  for  instance,  it 
resolves  A  into  two  distinct  pieces  (IV.),  and  that  one  of  these 
pieces  which  is  contiguous  to  Q,  together  with  B,  forms  one 
piece  of  T,  and  the  other  forms  a  second  piece  distinct  from 
the  former.  If,  however,  a  cross-cut  Q'  cross  Q  one  or  more 
times,  it  is  divided  by  the  points  of  intersection  into  parts 
which  form  cross-cuts  in  either  A  or  B,  and  which  therefore 
again  resolve  these  portions  into  distinct  pieces  (IV.).  Thus 
we  cannot  come  from  the  one  side  of  Q'  to  the  other  side 
either  in  A  or  in  B.  But  then  this  is  also  not  possible  in  T, 
i.e.,  by  crossing  Q,  because  thereby  we  always  come  only  from 
A  to  B,  or  conversely.  Therefore  Q'  likewise  divides  the  sur- 
face. Since  this  is  resolved  into  two  distinct  pieces  by  every 
cross-cut,  it  is  simply  connected  (VI.). 

VIII.  If  a  multiply  connected  surface  he  resolved  into  two 
distinct  pieces  hy  a  cross-cut,  at  least  one  of  the  pieces  is  again 
multiply  connected. 


196  THEORY  OF  FUNCTIONS. 

For,  if  both  were  simply  connected,  T  could  not  be  multiply 
connected  (VII.). 

IX.  If  a  surface  consisting  of  one  piece  be  resolved  by  q  cross- 
cuts into  q  -\-l  pieces,  each  of  which  is  by  itself  simply  connected, 
then  it  is  itself  simply  connected. 

Each  of  the  cross-cuts  drawn  divides  the  part  in  which  it 
falls  into  two  distinct  pieces ;  for,  if  only  a  single  one  should 
not  do  this,  there  would  at  the  end  be  less  than  g  +  1  distinct 
pieces,  since  a  cross-cut  can  never  divide  a  portion  into  more 
than  two  pieces  (III.).  If  the  given  surface  were  multiply 
connected,  the  first  cross-cut  could  at  most  cut  off  one  simply 
connected  piece  (VIII.),  the  other  piece  remaining  multiply  con- 
nected. If  we  now  assume,  in  order  to  emphasize  the  most 
favorable  case,  that  the  cross-cut  is  drawn  every  time  in  the 
multiply  connected  portion,  so  that  from  this  a  simply  con- 
nected piece  is  detached,  at  the  end  a  piece  would  remain  which 
is  not  simply  connected.  In  general,  by  each  mode  of  resolu- 
tion, either  less  than  q  +  1  pieces  would  have  been  formed,  or 
at  least  one  of  these  pieces  would  necessarily  be  multiply 
connected. 

49.  From  these  preliminary  principles  we  now  proceed  to 
the  following  fundamental  proposition :  — 

If  a  surface,  or  a  system  of  surfaces,  T,  be  resolved  in  one  way 
by  qi  cross-cuts  Qi  into  «i  distinct  pieces,  and  in  a  second  way  by 
q^  cross-cuts  Q2  into  etj  distinct  pieces,  in  such  a  manner  that  both 
the  «!  pieces  of  the  first  way  and  also  the  a^  pieces  of  the  second 
way  are,  each  by  itself,  simply  connected,  then  in  all  cases 

Qi  —  «i  =  92  —  «2- 

Proofs  —  The  two  systems  of  surfaces  formed  from  T  by 
means  of  the  cross-cuts  Qi  and  Q^  may  be  called  Ti  and  Tj 
respectively.  If  we  draw  either  the  lines  Q2  in  Ti,  or  the  lines 
Qi  in  T2,  we  obtain  in  both  cases  the  same  system  of  surfaces, 
exactly  the  same  figure.     Call  this  new  system  of  surfaces  ^. 

1  Riemann,  Grundlagen,  u.  s.  w.,  s.  6. 


SIMPLY  AND  MULTIPLY  CONNECTED   SURFACES.      197 

The  lines  Q^  form,  it  is  true,  q^  cross-cuts  ia  the  original 
surface  T,  but  not  necessarily  the  same  number  when  drawn 
in  Ti ;  for,  since  on  the  one  hand  the  lines  Q.,  cease  to  exist  as 
cross-cuts  in  7\  if  they  coincide  entirely  with  the  lines  Q„  and 
since  on  the  other  hand  they  may  be  divided  by  the  lines  Qi 
into  several  parts  (each  part  forming  a  distinct  cross-cut),  the 
number  of  cross-cuts  actually  formed  in  2\  by  the  lines  Q.,  may 
be  less,  or  even  greater  than  q.^.  Likewise,  also,  the  number 
of  cross-cuts  formed  by  the  lines  Qi  in  the  system  T2  may 
be  different  from  q^.  We  will  designate  the  cross-cuts  formed 
by  the  lines  Qo  in  T^  by  Q.,',  their  number  by  q.2';  the  cross-cuts 
formed  by  the  lines  Qi  in  T2  by  Qi,  their  number  by  g/.  The 
essential  feature  of  the  proof  consists  then  in  this,  that,  if 
we  let 

92'  =  92  +  m, 

then  also  q^'  z=  q^  -\-  m. 

To  prove  this,  let  us  direct  our  attention  to  the  end-points^ 
of  the  cross-cuts,  observing  that  the  number  of  cross-cuts  is 
half  as  great  as  the  number  of  their  end-points,  and  that  this 
is  invariably  the  case  if  we  only  count  twice  a  point  at  which 
the  initial  point  of  one  cross-cut  coincides  with  the  terminal 
point  of  another.  Accordingly,  the  number  of  end-points  of 
the  ^2  cross-cuts  Q.,  is  2  ^2-  But  if  these  be  regarded  as  cross- 
cuts Q2'  in  the  system  Tj,  already  resolved  by  the  lines  Qi,  on 
the  one  hand  some  end-points  of  the  lines  Qo  may  cease  to  be 
end-points  of  the  lines  Qo',  and  on  the  other  hand  new  points 
may  occur  as  end-points  of  the  lines  Qo'.  (Cf.  Fig.  47,  wherein 
the  lines  Qi  are  represented  by  the  heavier  lines,  the  lines  Q2 
by  the  lighter.  In  places  where  a  line  Qi  coincides  with  a 
line  Q2,  wholly  or  in  part,  they  are  represented  running  closely 
beside  each  other.) 

(1)  An  end-point  of  a  line  Q2  is  always  at  the  same  time  an 
end-point  of  a  line  Q2',  if  it  do  not  fall  on  one  of  the  lines  Qi 
(e.g.,  a  or  ^),  and  also  in  the  case  when  only  an  end-^om^  of  a 

1  We  will  call  the  initial  and  tenmnal  points  of  a  cross-cut  together  the 
two  end-points  of  the  same. 


198 


THEORY  OF  FUNCTIONS. 


line  Q2  coincides  with  an  end-point  of  a  line  Qi  (e.g.,  c  or  g). 
On  the  other  hand  an  end-pom^  of  a  line  Q2  is  not  at  the  same 
time  also  an  end-point  of  a  line  Q2,  if  the  line  Q^  coincide  for 
a  distance  from  this  end-point  (or  also  completely)  with  a  line 
Qi  (e.g.,  (Is,  8r,  op,  po).  In  such  a  case  the  point  in  question 
either  ceases  altogether  to  exist  as  an  end-point  of  a  line  Q,' 
(e.g.,  o  or  p),  or  it  is  only  displaced  as  such.  (While,  for  in- 
stance, the  cross-cut  ds/8,  regarded  as  a  line  Q2,  begins  at  d, 
the  same,  regarded  as  a  line  Q2,  begins  only  at  s ;  c?  is  there- 
fore an  end-point  of  a  line  Q^,  but  not  of  a  line  Q2.)  Kow 
this  can  occur  in  two  cases :  either  the  segments  which  coin- 
cide are  both  end-pieces  of  a  line  Q^  and  a  line  Q^  respec- 
tively (e.g.,  ds),  or  an  end-piece  of  a  line  Q2  coincides  with  a 


Fig.  47. 

mid-piece  of  a  line  Qi  (e.g.,  Sr,  op,  po).  If,  therefore,  v  be  the 
number  of  times  that  an  end-piece  of  a  line  Q2  coincides  with 
an  end-piece  of  a  line  Qj,  and  V2  be  the  number  of  times  that 
an  end-piece  of  a  line  Q2  coincides  with  a  mid-piece  of  a  line 
Qi,  then 

V  -f-  V2 

is  the  number  of  end-points  of  the  lines  Q2  whicli  are  not  at  the 
same  time  end-points  of  the  lines  Q2.  The  number  2  q2  of  end- 
points  of  the  lines  Q2  must  therefore  be  diminished  by  v  -f  V2. 


SIMPLY  AND  MULTIPLY  CONNECTED  SUBFACES.     199 

Similar  considerations  are  applicable  to  the  cross-cuts  Qj. 
An  end-point  of  a  line  Qj  ceases  to  be  an  end-point  of  a  line 
Qi,  when  an  end-piece  of  a  line  Qi  coincides  either  with  an  end- 
piece  of  a  line  Q2  (e.g.,  ds)  or  with  a  mid-piece  of  a  line  Q2  (e.g., 
eq).  If  therefore  vi  be  the  number  of  times  that  an  end- 
piece  of  a  line  Qi  coincides  with  a  mid-piece  of  a  line  Q2,  then 

V  +  Vi 

is  the  number  of  end-points  of  the  lines  Qi  which  are  not  at 
the  same  time  end-points  of  the  lines  Qi';  therefore  the  num- 
ber 2  Qi  of  end-points  of  the  lines  Qi  must  be  diminished  by 

(2)  But  now  new  points  appear  as  end-points  of  the  lines 
Qo'  or  Qi,  which  are  not  end-points  of  the  lines  Q2  or  Qi.  Let 
us  again  consider  first  the  lines  Q2.  As  new  end-points  of  these 
lines  appear,  in  the  first  place,  the  displaced  points  mentioned 
above  (e.g.,  r  or  s) ;  then,  also,  both  those  points  at  which  a 
mid-point  of  a  line  Q2  coincides  with  a  mid-point  of  a  line  Q^ 
(e.g.,  y  and  rf),  and  those  near  which  a  mid-piece  of  a  line  Q2 
coincides  with  a  mid-piece  of  a  line  Qi  (e.g.,  tii).  All  these 
cases  can  be  characterized  as  those  in  which  the  lines  Qi  and 
Q2  either  meet  or  separate  in  their  mid-course.  Let  fi  denote 
the  number  of  times  that  this  occurs.  The  following  consider- 
ations are  to  be  noted,  however,  concerning  the  determination 
of  this  number  /x.  In  the  first  place,  wherever  a  line  Q2  has 
common  with  a  line  Qi  only  a  single  mid-point  (not  a  segment ; 
e.g.,  at  y  or  rj),  this  point  must  be  counted  twice,  because  it  is 
a  terminal  point  of  a  line  Q2'  and  at  the  same  time  an  initial 
point  of  a  new  line  Q2'.  We  will,  however,  stipulate  that  the 
number  /x  be  so  determined  that  its  value  shall  be  independent 
of  whether  Ave  put  the  lines  Q2  in  relation  to  the  lines  Q„  or, 
conversely,  the  lines  Qi  in  relation  to  the  lines  Q,-  This 
requires  us  to  take  the  greater  number,  whenever  the  one 
relation  produces  a  greater  number  of  points  to  be  coimted 
than  the  other,  for  two  particular  cross-cuts.  The  points  which 
are  thus  counted  too  often  must  then  be  set  aside.  Now  this 
case  occurs  with  the  cross-cuts  Q2'  when,  and  only  when,  an  end- 


200 


THEOBY  OF  FUNCTIONS. 


piece  of  a  line  Qi,  coinciding  with  a  line  Q2,  terminates  ^  in  that 
mid-point  of  the  line  Q2  with  which  a  mid-point  of  another  line 
Qi  coincides  (e.g.,  at  e,  where  fe  ends ;  here  in  the  determination 
of  fi,  e  must  be  counted  twice,  because  xc  and  ab  have  only  one 
point  common  at  e,  and  e  is  at  the  same  time  an  end-point  of  ea 
and  em ;  but  e  occurs  on  xc  only  once  as  an  end-point,  namely, 
of  xe,  while  the  segment  ec,  regarded  as  a  line  Q2,  begins  only 
at  q,  which  point  must  likewise  be  counted  in  the  determina- 
tion of  fx).  This  case  occurs  therefore  when,  and  always  when, 
an  end-piece  of  a  line  Qi  coincides  with  a  mid-piece  of  a  line 
Q2 ;  for,  that  this  may  be  possible,  another  line  Qj-  must  at  the 
same  time  pass  through  the  end-point  of  this  line  Qj.  The 
number  of  times  that  an  end-piece  of  a  line  Qi  coincides  with  a 
mid-piece  of  a  line  Q2  was  denoted  above  by  vi.  Therefore, 
among  the  points  counted  in  the  determination  of  the  number 
fi,  are  vi  such  points  as  are  not  end-points  of  the  lines  Q2. 
Therefore  the  number  of  new  end-points  of  the  lines  Q2  which 
occur  amounts  to  „ 

iThis  case  can  also  occur  when  the  line  Qi  ends  in  a  point  of  its 
previous  course.     Cf.  the  third  note. 

2  Cf.  preceding  note. 

3  There  are  also  cases  in  which  the  number  n  can  be  computed  in 
different  ways  ;  the  difference  /jl  —  vi,  however,  remains  the  same.  In 
Fig.  48,  two  such  cases  are  given.  The  cross-cut  abcdb  (a  line  Qi),  end- 
ing in  a  point  of  its  previous  course,  coincides  with  ef  (as  a  line  Q2)  along 
the  part  bd.    Now,  if  we  take  the  first  in  the  sense  abdcb,  we  have  two 

points  fj.  on  each  cross-cut,  namely,  b  and 
d  ;  therefore  m  =  2,  and,  bd  being  at  the 
same  time  a  mid-piece,  vi  =  0.  If,  on 
the  other  hand,  we  take  the  line  Qi  in  the 
sense  abcdb,  b  is  to  be  counted  twice,  and 
we  have  therefore  now  /m  —  S  ;  at  the  same 
time,  however,  db  is  an  end-piece  of  the 
line  Qi  which  coincides  with  a  mid-piece 
of  the  line  Q2,  and  therefore  fi  =  1.  The 
difference  fi  —  viis  in  both  cases  the  same. 
The  other  example  is  similar  to  the  pre- 
ceding. In  this  we  have  the  choice  of 
assuming  Ihik  as  the  previous  cross-cut 


SIMPLY  AND  MULTIPLY  CONNECTED   SURFACES.     201 

The  reasoning  is  similar  in  the  determination  of  the  number 
of  points  which  enter  as  new  end-points  of  the  lines  Qi'.  The 
number  fx.  remains  the  same  as  before,  if  it  be  determined  in  the 
manner  given  above.  Among  these  fi  points,  however,  those 
are  not  end-points  of  the  lines  Q'l  near  which  an  end-piece  of 
a  line  Qo  coincides  with  a  mid-piece  of  a  line  Qi  (e.g.,  near  8, 
which  point  is  to  be  counted  twice ;  it  enters,  it  is  true,  as  an 
end-point  of  Sn,  but  not  of  8b,  because  this  cross-cut,  regarded 
as  a  line  Qi',  begins  only  at  r).  According  to  the  above  nota- 
tion this  occurs  V2  times ;  therefore  the  number  of  points  which 
enter  as  new  end-points  of  the  cross-cuts  Qi  equals 

/i.  —  Vg. 

Now,  according  to  (1)',  v  -f  V2  end-points  are  to  be  subtracted 
from  the  original  2  g,  end-points,  and,  according  to  (2),  /x  —  vi 
new  end-points  are  to  be  added.  Therefore  the  number  2  q^'  of 
end-points  of  the  cross-cuts  Q2  is 

2  ga' =  2  ga  -  (v  +  V2) -h  O  -  vi), 

t  t  I  .A*  —  (v  -f-  Vi  -f-  Vo) 

and  hence  gg  =  g2  +  - — ^ — ^ -' 

On  the  other  hand,  according  to  (1),  v  +  vi  end-points  are 
to  be  subtracted  from  the  original  2gi  end-points  of  Q^,  and 
according  to  (2),  /*  —  vg  are  to  be  added.  The  number  2  qi  of 
end-points  of  the  cross-cuts  Q/  therefore  is 

2gi'  =  2gi-(v-f  vi)  +  (fi-v2), 

and  hence  q^'  =  qi-{-  ^ — ^^ — —^ ^. 

and  hfg  as  the  subsequent  one,  or  gfhik  as  the  previous  and  hi  as  the 
subsequent  cross-cut.  The  piece  fhi  coincides  with  the  cross-cut  m7i  of 
the  second  kind.  If  we  choose  the  first  order,  we  have  to  count  three 
points,  h,  i,  and  /;  therefore  fi  =  3.  At  the  same  time,  however,  hf  is 
an  end-piece,  and  therefore  vi  —  1.  On  the  other  hand,  with  the  second 
order,  only/  and  i  are  to  be  counted,  and  therefore  m  =  2 ;  but  at  the 
same  time  i-^  =  0,  because  /fe  is  now  a  mid-piece. 


202  THEORY  OF  FUNCTIONS. 

Therefore,  if  we  let 

/x-(v  + Vi  +  V2) 

2  ~^' 

we  have  simultaneously 

q2=<l2  +  vri  and  gi'  =  5'i  +  m; 

this  result  was  first  to  be  proved. 

Let  us,  before  proceeding  further,  consider  Fig.  47  in  detail 
in  reference  to  the  above-described  relations. 


h     £ L 


In  the  following  table  the  cross-cuts  Qi  are  enumerated  in 
the  first  column,  and,  in  the  same  row  with  each,  are  given 
in  the  second  column  the  pieces  into  which  it  is  divided  if 
the  surface  be  regarded  as  previously  resolved  by  the  lines 
Q2;  the  latter  therefore  are  the  cross-cuts  Q/.  The  columns 
headed  Q^  ^^^  Q2  have  similar  meanings. 


SIMPLY  AND  MULTIPLY  CONNECTED   SURFACES.     203 


Qi- 

«.'. 

Q-2- 

Q-2'. 

ab 

ae,  em,  mw,  nS,  rb 

zc 

X€,    qc 

cd 

eg,  gy,  ys 

ez 

eij,     7)0,    oz 

ef 

qf 

xp 

xm,  mg,  gp 

gh 

gv,  vh 

yg 

yn,   ng 

ik 

to,  pk 

y/3 

yd,    dy,    7/3 

kl 

ku,  tl 

5v 

rt> 

Pd 

/3s 

ea 

et,     up,  pa 

op 

From  this  we  obtain 

5,  =  6,        qi'  =  15, 
q,  =  9,        qoJ^lS, 


m  =  9. 


'Let  us  fiirtlier  enumerate  the  points  fi,  indicating  each  point 
which  is  to  be  counted  twice  by  a  2  set  over  it : 


2    2     2    2          2    2 

2 

2    2 

e     m     n     S     r    g     y     s     q 

V 

0    p     < 

hence                                      /x  =  23. 

The  end-pieces  which  coincide  are : 

class  of  v'ds, 

v  =  l, 

"     "   vi  —  eq, 

VI  =  1, 

"      '•   v.2---8r,  o]),  po, 

V2  =  3; 

therefore          ^ -^^  "("  +  "^ +''^)  - 

23- 

9 

^  =  9. 

as  above. 

The  rest  of  the  proof  is  now  very  easily  given.  According 
to  the  hypothesis,  the  system  7\  consists  of  «i  distinct  pieces, 
each  by  itself  simply  connected.     From  this  the  system  ^  is 


204  THEORY  OF  FUNCTIONS. 

produced  by  the  q^  =  Q2  +  '"'-  cross-cuts  Q2.  The  latter  system 
consists  of  only  simply  connected  pieces  (§  48,  IV.).  Letting 
^  denote  the  number  of  the  latter,  it  follows  from  §  48,  V.  that 

Wi  =  Ui -\-  q., -\- m. 

The  same  system  ^  is  also  produced  from  To  by  the 

qi  =qi  +  m 

cross-cuts  Qi'.  Since,  however,  according  to  the  hypothesis, 
To  consists  of  «2  distinct  pieces,  each  by  itself  simply  con- 
nected, it  follows  also  that 

S  =  «2  +  Q'l  +  *«" 

Therefore,  «i  +  Q'2  +  ^'^  =  «2  +  Q*!  +  wi, 

or  q.2  —  a.2  =  ^1  —  «i. 

Note.  —  Having  prescribed  the  conditions  to  be  kept  in  mind  in  the 
proof  of  this  proposition,  we  can  prove  it  more  briefly  in  the  following 
manner  :  1  Let  us  assume,  in  the  first  place,  that  the  lines  Qi  and  Q^  have 
in  common  only  this,  that  a  line  of  the  one  kind  simply  crosses  one  of  the 
other,  and  in  such  a  way  that  the  point  of  intersection  is  not  an  end-point 
of  a  cross-cut.  If  such  a  case  occur  k  times,  we  have,  according  to  the 
above  explanation  and  with  the  preceding  notation, 

2q2'  =  2q2  +  2k,   2qi'  =  2qi  +  2k, 

and  therefore  q^'  =  q2  +  k,  qi'  =  qi  +  k. 

If,  however,  both  systems  lie  in  any  arbitrary  relation  to  each  other, 
the  lines  Qi  and  Q2  in  part  crossing  one  another  at  arbitrary  points,  in 
part  touching  one  another,  or  even  coinciding  with  one  another  wholly  or 
in  part,  then  an  infinitely  small  deformation  of  the  lines  of  one  system 
can  cause  the  coincidence  either  to  be  entirely  removed  or  to  conform 
only  to  the  foregoing  hypothesis.  If,  after  such  a  deformation,  k  points 
of  intersection  occur,  it  follows  again  that 

52'  =  q2  +  k  and  31'  =  qi  +  k, 

from  which  the  proposition  follows  as  above.  If  this  hold  after  the 
infinitely  small  deformation,  it  must  also  be  valid  before  the  same ;  for 
by  this  deformation  neither  the  number  of  cross-cuts  nor  the  number  of 
pieces  into  which  the  surface  is  resolved  is  changed. 

1  Neumann,  Vorlesungen  iiber  Biemann''s  Theorie,  u.  s.  w.,  S.  296. 


SIMPLY  AND  MULTIPLY  CONNECTED   SURFACES.     205 

50.  The  proof  of  the  fundamental  proposition  given  by 
Biemann  and  set  forth  in  the  preceding  paragraph  renders 
detailed  discussions  necessary,  if  none  of  the  cases  which 
might  possibly  occur  are  to  be  overlooked. 

Neumann's  proof  is  shorter,  it  is  true,  but  it  does  not  make 
use  of  the  property  Avhich  forms  the  principal  part  of  the 
preceding  proof,  that  in  general 

This  property  is  important  in  itself,  however,  and  will  later 
have  to  be  applied.  On  that  account  it  is,  perhaps,  not  un- 
profitable to  add  still  another  proof  of  the  fundamental  propo- 
sition, which  is  simpler  and  which  yet  employs  the  above 
property.  Such  a  proof  has  been  given  by  Lippich.^  It  pre- 
supposes, it  is  true,  some  knowledge  of  the  properties  of 
general  line-systems,  but  it  is  characterized  by  great  simplicity 
and  leads,  moreover,  to  a  new  and  very  important  principle. 
To  give  this  proof,  we  must  introduce  the  following  preliminary 
considerations.^ 

Digression  on  line-systems.  We  will  consider  a  system  of 
straight  or  curved  lines  which  in  other  respects  can  be  quite 
arbitrary ;  yet  we  do  assume  that  no  line  of  the  system  ex- 
tends to  infinity,  and  also  that  none  makes  an  infinite  num- 
ber of  windings  in  a  finite  region.  We  further  assume  that 
all  the  lines  are  connected  with  one  another,  or,  if  this  be  not 
the  case,  that  they  form  only  a  finite  number  of  pieces. 

From  each  point  of  a  line  we  can,  following  one  or  more 
lines,  take  either  a  single  path  or  several  paths.  Accordingly 
we  distinguish  three  kinds  of  points  on  the  lines :  (1)  Let  a 
point  from  which  we  can  take  only  one  path  be  called  an  end- 
point;  e.g.,  a,  a,,  aa,  a^,  in  Fig.  49.  (2)  Call  a  point  from  which 
we  can  proceed  by  two  paths  an  ordinary  point;  e.g.,  b,  6],  62- 
(3)  Let  a  point  from  which  we  can  take  more  than  two  paths 

1  F.  Lippich,  "Bemerkung  zu  einem  Satze  aus  Kiemann's  Theorie  der 
Funktionen,"  u.  s.  w.  (Sitz.-Ber.  d.  Wien.  Acad.,  Bd.  69,  Abth.  II., 
Januar  1874.) 

2  The  continuation  of  the  main  investigation  follows  in  §  52. 


206 


THEORY  OF  FUNCTIONS. 


Fig.  49. 


be  called  a  nodal^oint,  and  let  it  be  called  w-ple,  when  n  paths 
proceed  from  it,  i.e.,  when  n  line-segments  meet  in  it.     In  Fig. 

49,  c  is  a  triple,  d  a  quadruple, 
e  a  quintuple  nodal-point.  Now 
if  a  line-system,  constituted  in 
this  way,  contain  either  no  end- 
and  nodal-points,  or  only  a  finite 
number  of  such  points,  it  will  be 
called  a  Jinite  line-system.  We 
will  introduce  the  following  addi- 
tional definitions.  Let  a  contin- 
uous line,  which  contains  two 
end-points  and  in  addition  only  ordinary  points,  be  called  a 
simple  line-segment,  e.g.,  ahcii ;  let  a  continuous  line  which  con- 
tains only  ordinary  points  be  called  a  simply  dosed  line,  e.g.,  A. 
If,  as  will  immediately  occur,  a  description  of  connected 
lines  of  the  system  be  under  discussion,  it  will  always  be 
assumed  that  therein  no  line-segment  is  described  more  than 
once,  but  that  the  description  is  always  continued  and  does 
not  cease  as  long  as  a  line-segment,  not  yet  described  and 
connected  with  the  segment  last  traced,  presents  a  path  for 
further  motion.  Then,  on  account  of  the  hypothesis  made  in 
regard  to  the  construction  of  the  line-system,  the  description 
must  always  come  to  an  end  sometime.  This  always  occurs 
on  arriving  at  an  end-point,  but  at  an  ordinary  point  when, 
and  only  when,  this  was  at  the  same  time  the  initial  point. 
At  all  the  nodal-points,  sections  are  always  to  be  associated 
with  the  description  in  such  a  way  that  when  a  nodal-point  is 
passed  for  the  first  time,  when,  therefore,  we  arrive  at  the 
nodal-point  on  one  line-segment  and  leave  it  on  any  other,  all 
the  other  line-segments  meeting  there  are  to  be  regarded  as 
cut.  Each  of  these  latter,  then,  acquires  an  end-point  at  the 
nodal-point,  and  its  connection  with  the  other  lines  is  regarded 
as  broken  at  that  point.  If  this  method  be  adhered  to,  the 
description  does  not  stop  at  a  nodal-point  when  we -arrive  at  it 
for  the  first  time,  but  always  ends  when  we  return  to  it  for 
the  second  time. 


SIMPLY  AND  MULTIPLY  CONNECTED   SURFACES,     207 

If  a  line-system  do  not  contain  any  nodal-points,  a  single 
end-point  cannot  occur  in  it;  the  end-points,  provided  there 
are  any,  must  rather  occur  in  pairs.  For  if  the  stated  descrip- 
tion begin  at  one  of  the  existing  end-points,  it  can  end  neither 
in  a  nodal-point  (since  there  are  none),  nor  in  an  ordinary 
point  (since  the  initial  point  was  not  an  ordinary  point)  ;  thus 
it  can  end  only  in  an  end-point,  and,  since  an  end  of  the 
description  must  sometime  occur,  there  must  still  be  a  second 
end-point.  Therefore,  starting  from  an  end-point,  we  always 
arrive  at  another  end-point,  and,  since  in  the  meantime  we 
have  passed  through  only  ordinary  points,  we  have  traced  a 
simple  line-segment.  Each  existing  end-point,  therefore,  in  a 
finite  line-system  containing  no  nodal-points,  is  associated  with 
a  second  end-point,  the  two  together  bounding  a  simple  line- 
segment.  Since,  now,  conversely,  every  simple  line-segment 
also  possesses  two  end-points,  we  conclude :  If  a  finite  line- 
system  contain  no  nodal-points,  the  number  of  simple  line-segments 
constituting  it  is  half  as  great  as  the  number  of  existing  end- 
points. 

In  the  case  of  an  arbitrary  finite  line-system  affected  with 
nodal-points,  we  can  remove  the  nodal-points  altogether  by 
means  of  the  sections  mentioned  above,  by  regarding  all  the 
line-segments  meeting  at  each  nodal-point  as  cut  except  two, 
which  are  to  be  left  connected.  Then  each  nodal-point  is 
changed  into  an  ordinary  point  and  a  number  of  end-points ; 
and  in  fact,  if  the  point  be  an  /i-ple  nodal-point,  it  is  changed 
into  one  ordinary  point  and  h  —  2  end-points.  Now,  since  the 
two  line-segments  left  in  connection  at  each  nodal-point  can 
be  chosen  quite  arbitrarily,  the  sections  may  be  effected  in 
very  many  different  ways.  These  having  been  effected,  the 
line-system  no  longer  contains  nodal-points,  and  therefore  the 
simple  line-segments  then  contained  in  it  can  be  found  accord- 
ing to  the  preceding  principle,  by  counting  the  number  of  end- 
points  which  occur.  For  instance,  if  we  denote  the  number  of 
end-points  contained  in  the  original  system  by  e,  the  number 
of  triple,  quadruple,  •••,  w-ple  nodal-points  by  kg,  A:^,  •••,  A;„,  we 


208  THEORY  OF  FUNCTIONS. 

have,  since  each  /t-ple  nodal-point  furnishes  h  —  2  end-points, 
in  all 

simple  end-points ;  therefore  the  number  of  simple  line-segments 
occurring  after  the  sections  are  effected  equals 

^[e  +  k,  +  2k,  +  3k,+...+  {n-2)Jc„y 

This  number  depends,  however,  upon  the  originally  existing 
ead-  and  nodal-points,  and  is  entirely  independent  of  the  way 
in  which  the  sections  were  effected,  which,  as  stated,  can  be 
made  in  very  many  different  ways.  We  therefore  conclude : 
If  all  existing  nodal-points  be  removed  from  a  given  finite  liner- 
system,  after  the  sections  are  effected  the  system  contains  a  series 
of  simple  line-segments,  the  number  of  which  is  constant  and  inde- 
pendent of  the  way  in  which  the  sections  were  effected. 

The  line-system  may  originally  contain  simply  closed  lines, 
but  the  sections  can  always  be  effected  in  such  a  way  that 
thereby  no  new  simply  closed  lines  are  furnished.  For  this 
could  only  occur  when,  after  all  line-segments  except  three 
have  been  cut  at  a  nodal-point,  a  particular  one  of  these  three 
is  then  cut.  But  if  we  choose  to  cut  one  of  the  two  other  line- 
segments  instead  of  that  particular  one,  no  simply  closed  line 
will  thereby  be  furnished.  If  the  sections  be  effected  in  this 
way,  and  if  the  system  contain  originally  no  simply  closed 

^  The  number 

e -J- *3  +  2  *4 -1- 3  A;6 -f  •••  + (w  -  2)fc„ 

is,  according  to  the  above,  an  even  number.  If  we  take  from  it  first  the 
even  number 

2^-4-|-4A;6-l-— , 
and  then  the  even  number 

2A:5  -l-4i-7  +  -.-, 
the  remaining  number 

e -I-  ^3 -I-  A;5  +  AtH — 

must  also  be  even.  Thus  it  follows  that:  In  a  finite  line-system  the  num- 
ber of  end-points  and  of  odd  nodal-points  together  is  always  an  even 
mtmber.  For  instance,  it  is  not  possible  to  draw  a  line-system  containing, 
say,  two  end-points  and  a  quintuple  nodal-point  (and  no  more  nodal- 
points). 


SIMPLY  AND  MULTIPLY  CONNECTED   SURFACES.     209 

lines,  after  the  sections  are  effected  it  consists  of  only  simple 
line-segments  constant  in  number. 

It  is  of  advantage,  in  a  system  containing  no  simply  closed 
lines,  to  conceive  the  resolution  into  simple  line-segments  as 
effected  by  successive  descriptions,  with  which,  as  already 
stated,  sections  are  to  be  associated,  at  every  nodal-point 
passed.  Then  simply  closed  lines  can  never  occur,  not  even 
in  the  case  of  a  triple  nodal-point,  because  the  description 
stops  at  a  nodal-point  only  when  we  arrive  at  it  for  the 
second  time.  To  effect  the  resolution,  we  begin  the  description 
at  an  end-point;  then  we  must  once  more  arrive  at  an  end- 
point,  which  either  originally  existed  or  was  furnished  by  a 
section  at  a  nodal-point  previously  passed.  We  have  then 
traced  a  simple  line-segment.  We  next  begin  the  description 
again,  either  at  an  originally  existing  end-point  or  at  one 
furnished  by  a  section,  and  obtain  a  second  line-segment,  etc. 
Should  the  case  occur  that  there  is,  on  one  of  the  simple  line- 
segments  to  be  traced,  no  end-point  at  which  to  begin  the 
description,  then  there  must  be  a  nodal-point  in  some  part  of 
the  system  not  yet  described;  for,  otherwise,  this  portion 
would  contain  only  ordinary  points,  and  therefore  consist  of 
only  simply  closed  lines,  while,  according  to  the  hypothesis, 
the  system  does  not  contain  any  such  lines.  In  such  a  case 
we  get  an  end-point  at  a  nodal-point  by  cutting  only  one  line- 
segment,  and  begin  the  description  at  this  point.  In  doing 
so,  if  the  point  be  a  triple  nodal-point,  we  must  certainly  be 
careful  not  to  cut  exactly  that  line-segment  by  the  section  of 
which  a  simply  closed  line  could  be  furnished.  We  therefore 
obtain  the  following  proposition :  A  finite  line-system,  not  con- 
taining any  simply  closed  lines,  is  resolved  by  successive  descrip- 
tions (with  which  a  section  is  to  be  associated  at  every  nodal-point 
passed)  into  only  simple  line-segments,  the  number  of  which  is 
always  the  same  in  whatever  way  the  description,  and  with  it  the 
resolution  into  simple  line-segments,  may  be  effected. 

51.  AVe  will  now  assume  that  in  a  given  surface  there  is  a 
finite  line-system  L,  which  satisfies  the  two  following  conditions : 


210  THEORY  OF  FUNCTIONS. 

(1)  No  end-point  is  to  lie  in  the  interior  of  the  surface,  but 
all  end-points  which  occur  are  to  be  situated  on  the  boun- 
dary of  the  surface.  (If  the  surface  be  closed,  it  is  to  be 
regarded  as  having  a  boundary-point,  by  §  46.) 

(2)  All  parts  of  the  system  L  are  to  be  connected  with  the 
boundary  of  the  surface,  so  that,  by  following  the  lines,  we 
can  arrive  at  the  boundary  of  the  surface  from  any  point 
situated  on  L. 

Such  a  line-system  can,  it  is  true,  contain  simply  closed 
lines,  yet  every  simply  closed  line,  in  order  to  satisfy  con- 
dition (2),  must  have  at  least  one  point  in  common  with  a 
boundary-line. 

We  now  first  add  the  boundary-lines  to  the  system  L ;  then 
every  point  of  L  situated  on  the  boundary  is  a  nodal-point  of 
the  system  modified  by  the  addition  of  the  boundary-lines, 
since  there  meet  in  it  at  least  three  line-segments,  namely, 
two  belonging  to  the  boundary-line,  and  at  least  one  belonging 
to  the  system  L.  The  same  is  also  true  of  the  boundary-point 
assumed  in  a  closed  surface,  if  we  conceive  this  as  an  infi- 
nitely small  boundary-line.  Let  us  now  effect  the  sections 
at  all  these  nodal-points  on  the  boundary  in  such  a  way  that 
the  line  segments  belonging  to  the  boundary-line  are  left 
connected,  but  all  the  line-segments  belonging  to  the  system 
L  are  cut.  (Thus  all  the  line-segments  meeting  at  a  mere 
boundary-point  must  be  cut.)  If  we  next  exclude  again  the 
boundary-lines,  we  have  changed  the  system  L  into  another 
system,  which  we  may  designate  by  M.  Each  of  the  points 
situated  on  the  boundary  in  the  latter  system  is  now  an  end- 
point  ;  and  there  must  be  at  least  one  such  point,  if  the  system 
L  is  to  satisfy  condition  (2).  Further,  the  system  M  does 
not  contain  any  closed  lines,  since  those  which  may  have 
existed  in  L  have  been  removed  by  means  of  the  sections. 
In  other  respects,  however,  the  lines  of  M  are  the  same  as  the 
lines  of  L. 

Therefore,  according  to  the  last  proposition  of  the  preceding 
paragraph,  the  system  M  can  be  resolved  by  successive  descrip- 
tions into  none  but  simple  line-segments,  constant  in  number. 


SIMPLY  AND  MULTIPLY  CONNECTED   SURFACES.     211 

But  it  is  now  easy  to  show  that  these  line-segments  constitute 
a  system  of  cross-cuts  drawn  in  the  surface.  For,  since  from 
the  preceding  considerations  there  is  at  least  one  end-point 
situated  on  the  boundary,  the  first  line-segment  can  be  traced 
from  such  a  point.  The  second  end-point,  at  which  we  then 
arrive,  cannot  by  (1)  be  in  the  interior,  but  it  is  either  on  the 
boundary  or  it  has  been  furnished  by  a  section  at  a  nodal-point 
passed.  But  in  both  cases  the  simple  line-segment  traced  is 
a  cross-cut ;  in  the  second  case  it  is  such  a  cross-cut  as  ends 
in  a  point  of  its  previous  course.  IS^ow  there  must  be  still 
another  end-point  (in  case  there  are  any  lines  not  yet  traced), 
which  is  situated  either  on  the  boundary  or  on  the  first  cross- 
cut, since,  otherwise,  condition  (2)  would  not  be  satisfied. 
Thus  we  can  trace  a  second  simple  line-segment,  starting  from 
this  point;  the  second  end-point  of  this  segment  lies  either 
on  the  boundary  or  on  the  first  cross-cut,  or  it  is  furnished 
by  a  section  on  passing  through  a  nodal-point.  This  second 
line-segment  is  therefore  also  a  cross-cut.  The  same  process 
is  repeated  as  long  as  there  remain  lines  not  yet  traced ;  there- 
fore the  line-segments  collectively  do,  in  fact,  constitute  a 
system  of  cross-cuts,  and  the  following  proposition,  established 
by  Lippich,  follows :  Every  finite  line-system  which  occurs  in  a 
surface,  and  which  satisfies  conditions  (1)  and  (2),  forms  a  sys- 
tem of  cross-cuts,  completely  determined  in  mimher,  and  this 
number  remains  always  the  same  in  whatever  way  these  cross-cuts 
may  he  successively  drawn. 

From  the  definition  of  cross-cuts  (§  47)  it  is  immediately 
obvious  that,  conversely,  every  system  of  cross-cuts  forms  a 
line-system  which  satisfies  conditions  (1)  and  (2). 

The  proof  of  Riemann's  fimdamental  proposition,  based 
upon  this  principle,  is  very  simple.  In  it  we  retain  the  nota- 
tion of  §  49.  If  we  suppose  first  the  lines  Qi  and  then  the 
lines  Qa  drawn  in  the  surface  T,  we  obtain  a  line-system  which 
consists  only  of  cross-cuts,  and  which  therefore  satisfies  con- 
ditions (1)  and  (2).  According  to  the  preceding  proposition 
this  now  forms  at  the  same  time  a  system  of  cross-cuts,  fully 
determined  in  number,  and  this  number  may  be  denoted  by  s. 


212  THEORY  OF  FUNCTIONS. 

But  the  number  of  the  cross-cuts  Qi  was  q^;  if  these  be  drawn, 
the  lines  Q2  form  q^'  new  cross-cuts,  using  the  former  notation, 
and  the  total  number  of  cross-cuts  is  thus  gi  -f  g'2' ;  therefore 

If  now,  conversely,  the  cross-cuts  Q2  be  first  drawn,  the  num- 
ber of  which  was  q.^,  and  then  the  lines  Qi  be  added,  forming 
Q'l'  new  cross-cuts,  then  in  all  §'2  +  5'i'  cross-cuts  are  obtained. 
But  the  line-system  which  now  exists  is  exactly  the  same  as 
before,  except  that  the  cross-cuts  of  which  it  consists  have 
been  drawn  in  a  different  way.  Since  according  to  the  above 
proposition  the  number  of  these  cross-cuts  must  nevertheless 
be  the  same,  it  follows  that 

g2  +  gi'  =  s; 

therefore,     q^  +  q^!  =  go  +  9i'  or  q<l  —  q^  —  g/  —  gj. 

If  we  denote  the  common  value  of  these  differences  by  m,  we 
have 

Q''i  =  gi  +  m,  ga'  =  q>i-^m. 

Having  established  the  first  and  principal  part  of  the  proof, 
we  proceed  exactly  as  in  §  49. 

52.  Let  us  now  consider  the  case  in  which  the  original  sur- 
face T  consists  of  a  single  connected  piece,  and  in  which, 
further,  each  of  the  surfaces  T^  and  T^?  obtained  by  means  of 
the  cross-cuts  Qi  and  Q25  forms  a  single  simply  connected  sur- 
face. In  order  that  this  case  may  occur,  it  is  necessary  in  the 
first  place  that  none  of  the  cross-cuts  divide  the  surface ;  there- 
fore also,  by  §  48,  II.  and  IV.,  that  T  be  multiply  connected 
and  remain  multiply  connected,  for  both  modes  of  resolution, 
until  the  next  to  the  last  cross-cut  has  been  drawn,  and  be 
rendered  simply  connected  only  by  means  of  the  last  cross-cut. 
In  such  a  case  «,  =  a2  =  1 ;  accordingly  gg  —  1  =  gi  —  1,  and 
hence  q2=qi-  From  this  result  we  obtain  the  following  propo- 
sition : 


SIMPLY  AND  MULTIPLY  CONNECTED   SURFACES.     213 

If  it  he  possible  by  means  of  cross-cuts  to  modify  a  multiply 
connected  surface  into  one  simply  connected,  and  if  this  be  possi- 
ble in  more  than  one  way,  then  the  number  of  cross-cuts  by  means 
of  ivhich  the  modification  is  effected  is  always  the  same. 

In  addition  to  this  proposition  the  following  is  of  the  greatest 
importance : 

If  a  multiply  connected  surface  can  be  modified  into  one  simply 
connected  in  any  one  definite  way  by  means  of  q  cross-cuts,  then 
this  modification  is  always  effected  by  means  of  q  cross-cuts,  in 
whatever  way  these  may  be  drawn,  provided  only  that  they  do  not 
divide  the  surface. 

Although  it  has  been  proved  in  the  preceding  proposition 
that  the  number  of  cross-cuts  remains  the  same  if  the  resolu- 
tion into  a  simply  connected  surface  be  possible  in  a  second 
way,  yet  it  remains  to  be  discussed  whether  this  resolution  can 
in  fact  be  effected  in  a  second  way ;  whether,  on  the  contrary, 
for  the  modification  into  a  simply  connected  surface,  the 
cross-cuts  must  not  be  drawn  in  a  definite  way,  according  to 
a  definite  rule,  so  that,  if  they  be  not  so  drawn,  the  surface 
always  remains  multiply  connected  and  a  simply  connected 
surface  is  never  obtained,  however  far  the  drawing  of  the 
cross-cuts  may  be  continued.  But  we  can  in  fact  show  that 
this  case  cannot  occur.  We  therefore  assume  that  the  surface 
T  is  modified  by  q  cross-cuts  Qi,  drawn  in  a  definite  way,  into 
a  simply  connected  surface  Tj.  Then  in  the  first  place  it 
follows  from  the  preceding  proposition  that,  if  instead  of  the 
former  cross-cuts  others  be  drawn,  the  surface  T  cannot  be 
made  simply  connected  by  means  of  less  than  q  cross-cuts. 
Hence,  by  §  48,  II.,  it  is  possible  to  draw  q  other  cross-cuts 
Qo,  which  likewise  do  not  divide  the  surface,  by  means  of 
which  a  surface  T2  may  be  formed ;  the  question  then  arises, 
whether  T2  must  be  simply  connected.  Let  us  form,  as  in 
§  49,  a  new  system  of  surfaces  oT  from  7\  and  Tj  in  two  ways, 
first  by  drawing  the  lines  Q2  in  T^,  and  secondly  by  drawing 
the  lines  Qi  i^  ^2-  -^.s  in  §  49,  let  the  number  of  cross-cuts 
which  the  lines  Q2  form  in  2\  be  denoted  by  g  -f  m.     Then, 


214  THEOBY  OF  FUNCTIONS. 

according  to  the  first  part  of  tlie  proof  of  tlie  fundamental 
proposition  (§  49  or  §  51),  the  number  of  cross-cuts  which  the 
lines  Qi  form  in  T-j  is  likewise  q  -\-  m.  Now  Tj  is,  according 
to  the  assumption,  a  single  simply  connected  surface,  which  is 
transformed  by  g  +  m  cross-cuts  into  the  system  of  surfaces 
^ ;  therefore  ^  consists  of  g  +  m  -f  1  distinct  pieces,  each  by 
itself  simply  connected  (§  48,  V.).  But  the  same  system  is 
also  produced  from  T2  hy  q  +  m  cross-cuts ;  therefore  the  sur- 
face T.2,  which  consists  of  one  piece,  has  the  property  that  it 
is  resolved  by  g  +  m  cross-cuts  into  g  +  m  -f  1  distinct  pieces, 
each  by  itself  simply  connected.  Thus,  by  §  48,  IX.,  T^  is  in 
fact  simply  connected. 

Upon  this  is  based  a  classification  of  surfaces  and  the  more 
exact  determination  of  their  connection. 

If  a  surface  be  multiply  connected,  a  cross-cut  can  be  drawn 
in  it  which  does  not  divide  the  surface  (§  48,  II.).  If  the  case 
occur  that,  after  the  addition  of  this  first  cross-cut,  the  surface 
has  become  simply  connected,  then  according  to  the  last  propo- 
sition it  is  changed  into  a  simply  connected  surface  by  every 
cross-cut  which  does  jiot  divide  it.  In  this  case  the  surface 
is  said  to  be  doubly  connected. 

But  if,  after  the  first  cross-cut  is  drawn,  the  surface  remain 
multiply  connected,  a  new  cross-cut  can  be  drawn  which  does 
not  divide  it.  If  this  change  it  into  a  simply  connected  sur- 
face, the  same  result  is  obtained  by  means  of  any  other  two 
cross-cuts  which  do  not  divide  the  surface.  The  surface  is 
then  said  to  be  triply  connected. 

If  the  surface  be  still  multiply  connected  after  the  addition 
of  the  second  cross-cut,  a  third  can  then  be  drawn  which  does 
not  divide  the  surface,  and,  according  as  the  resolution  into  a 
simply  connected  surface  is  effected  by  means  of  three,  four, 
etc.,  cross-cuts,  the  surface  is  said  to  be  quadruply,  quintuply, 
etc.,  connected. 

In  general,  a  surface  is  said  to  be  (g  +  l)-ply  connected  when 
it  can  be  changed  by  means  of  q  cross-cuts  into  a  simply  connected 
surface.  In  that  case  it  is  unimportant  how  the  cross-cuts  are 
drawn,  provided  only  that  none  of  them  divide  the  surface. 


SIMPLY  AND  MULTIPLY  CONNECTED   SURFACES.     215 

But  after  the  surface  becomes  simply  connected,  it  is  no  longer 
possible  to  draw  a  cross-cut  in  it  which  does  not  divide  the 
surface  (§  48,  IV.). 

53.  We  are  now  prepared  to  prove  some  propositions,  in 
part  relating  to  the  variation  or  non-variation  of  the  order 
of  connection/  in  part  relating  to  boundary-lines. 

I.  Tlie  order  of  connection  of  a  surface  is  diminished  by  unity 
by  every  cross-cut  which  does  not  divide  it. 

For,  if  the  surface  be  {q  +  l)-ply  connected,  it  follows  from 
the  second  proposition  of  §  52  that,  however  the  first  cross-cut 
which  does  not  divide  the  surface  may  be  drawn,  the  resolution 
into  a  simply  connected  surface  is  always  effected  by  means 
of  g  —  1  cross-cuts;  hence  the  new  surface  is  9-ply  connected. 

II.  If  a  line  he  drawn  from  a  point  a  of  the  boundary  into  the 
interior  of  the  surface,  and  if  without  returning  into  itself,  it 
end  in  a  point  c  in  the  interior  of  the  surface,  such  a  line  does 
not  change  the  order  of  connection  of  the  surface.  (If  a  section 
be  made  along  this  line,  it  is  called  a  slit.) 

Call  the  original  surface  T  and  that  formed  by  the  intro- 
duction of  the  line  ac,  T'.  In  the  first  place  it  is  evident  that, 
if  T  be  simply  connected,  T'  must  also  be  simply  connected ; 
for,  if  every  closed  line  in  T  form  by  itself  alone  the  complete 
boundary  of  a  portion  of  the  surface,  so  also  does  every  closed 
line  in  T',  i.e.,  every  line  which  does  not  cross  ac.  Therefore 
let  T  be  multiply  connected,  say  {q  -f-  l)-ply.  Then  a  cross- 
cut which  does  not  divide  the  surface  can  ahvays  be  drawn  in 
T  (§  48,  II.),  and  in  fact  so  that  the  line  ac  forms  part  of  the 
same.  This  is  always  possible ;  for,  if  the  cross-cut  be  drawn 
as  directed  in  §  48,  II.,  with  the  help  of  a  closed  line  which 
does  not  form  by  itself  alone  a  complete  boundary,  it  can  be 
made  to  run  from  a  point  of  the  latter  on  each  side  to  the 
edge  of  the  surface  in  an  entirely  arbitrary  way;  therefore, 
since  a  lies  on  the  edge,  so  that  ac  always  forms  part  of  the 
same.  Let  this  cross-cut  be  denoted  by  acb,  and  let  the  surface 
formed  from  T  by  means  of  it  be  called  T";  then  the  latter 

1  Sometimes  called  connectivity.     (Tr.) 


216  THEORY  OF  FUNCTIONS. 

surface  is  5-ply  connected  (I.).  But  T"  can  also  be  formed 
from  T'  by  means  of  the  line  cb,  and  this  line  forms  in  T'  a 
cross-cut  which  does  not  divide  the  surface,  because  it  is  so 
drawn  that  the  contiguous  portions  of  the  surface  on  both 
sides  of  it  are  connected.  Consequently  T'  has  the  property, 
that  it  is  changed  into  a  Q'-ply  connected  surface  by  means  of 
one  cross-cut  which  does  not  divide  the  surface ;  therefore  T" 
is  (q  +  l)-ply  connected,  just  as  T  was. 

Note.  —  This  proposition  remains  perfectly  valid,  if  the  internal  point 
c  be  a  branch-point. 

III.  If  a  single  point  c  be  removed  from  a  surface  T  at  any 
place,  the  order  of  connection  is  thereby  increased  by  unity. 

Let  the  surface  formed  by  the  removal  of  the  point  c  be 
called  T'.  Connect  the  point  c  with  any  point  a  of  the  bound- 
ary of  T^  by  means  of  a  line  which  does  not  intersect  itself, 
thereby  forming  a  new  surface  T".  Then  the  latter  can  also 
be  regarded  as  a  surface  formed  from  T  by  drawing  the  line 
ac,  which  starts  from  a  boundary-point  a  and  ends  in  an 
interior  point  c ;  therefore  the  order  of  connection  of  T"  is  the 
same  as  that  of  T  (II.).  On  the  other  hand,  ac  is  a  cross-cut 
in  T"  which  does  not  divide  the  surface,  since  we  can  pass 
round  c  from  the  one  side  to  the  other.  Accordingly,  by  I., 
T'  is  of  an  order  higher  by  unity  than  T",  and  therefore  also 
than  T. 

Note.  — The  preceding  does  not  lose  its  validity  if  the  point  removed 
be  a  branch-point. 

IV.  If  an  (actually)  closed  line,  forming  by  itself  alone  the 
complete  boundary  of  a  portion  of  a  surface,  be  drawn  in  any 
position  in  a  portion  which  contains  either  no  branch-point  or  at 
most  one  (of  any  order  [§  13]),  and  if  the  portion  bounded  by 
this  line  be  removed  from  the  surface,  the  order  of  connection  is 
thereby  increased  by  unity. 

For  the  order  of  connection  will  not  be  changed,  if  the 
boundary-line  which  bounds  the  piece  removed  be  more  and 

1  If  y  be  a  closed  surface,  it  is  assumed  that  it  already  possesses  a 
boundary-point  a  (§40). 


SIMPLY  AND  MULTIPLY  CONNECTED   SURFACES.     217 

more  contracted.  But  if  it  finally  shrink  into  a  point,  the  case 
is  the  same  as  the  preceding.  Hence  this  proposition  holds, 
if  the  piece  removed  contain  either  no  branch-point  or  only 
one.  But  if  it  contain  more  than  one,  it  would  no  longer  be 
possible  to  let  the  boundary-line  shrink  into  a  point. 

Co)\  In  the  case  of  a  surface  closed  at  infinity,  it  is 
necessary  to  assume  in  some  place  a  boundary-point  (§  46). 
This  may  itself  also  be  a  branch-point.  If  a  piece  which  con- 
tains this  boundary-point,  hut  no  other  branch-point,  be  removed 
from  such  a  surface,  the  order  of  connection  is  not  changed.  For 
"we  can  let  the  boundary-line  which  bounds  the  piece  removed 
contract  into  the  boundary-point,  and  thus  obtain  again  the 
original  surface. 

V.  Ifa(q-\-  l)-2>ly  connected  surface  T  be  resolved  by  a  cross- 
cut R  into  two  distinct  pieces  A  and  B,  each  of  the  latter  has  a 
finite  order  of  connection  ;  and  if  r  and  s  be  the  numbers  of  cross- 
cuts which  determine  these  orders,  then  r  -\-  s  =  q. 

The  surface  T  is  reduced  to  two  simply  connected  pieces  by 
means  of  g  -f  1  cross-cuts,  of  which  the  first  q  do  not  divide 
the  surface.  If,  however,  the  dividing  cross-cut  E  be  first 
drawn,  we  cannot  immediately  infer  the  truth  of  the  above 
enunciation  from  Riemann's  fundamental  proposition,  because 
that  proposition  already  assumes  what  will  here  be  first  proved, 
namely,  that  now  also  after  the  addition  of  a  finite  number  of 
cross-cuts  simply  connected  pieces  are  finally  obtained  again. 
We  remark  that  all  the  cross-cuts  running  in  A  can  be  so 
placed  that  they  meet  the  cross-cut  R  either  not  at  all  or  only 
in  one  of  its  end-points.  For,  since  M,  except  for  its  end- 
points,  lies  entirely  in  the  interior  of  the  surface,  and  since 
therefore  a  zone  free  of  boundary-lines  exists  on  each  side  of 
it,  we  can  displace  along  the  line  M  the  end-points  of  all  the 
cross-cuts  which  meet  li  until  they  coincide  with  an  end-point 
of  the  same.  But  then  every  cross-cut  which  does  not  divide 
A  is  also  a  non-dividing  cross-cut  in  T.  From  this  it  follows 
that  the  number  r  of  non-dividing  cross-cuts  possible  in  A  can- 
not be  greater  than  q ;  for,  otherwise,  it  would  also  be  possible 
to  draw  in  T  more  than  q  non-dividing  cross-cuts,  and  this  is 


218  THEORY  OF  FUNCTIONS. 

contrary  to  the  hypothesis  that  this  surface  is  (q  +  l)-ply 
connected  (§  52).  Therefore  r  is  a  finite  number,  and  when 
these  r  cross-cuts  have  been  drawn  in  A,  another  non-dividing 
cross-cut  is  qo  longer  possible;  thus  A  has  become  simply 
connected  (§  48,  VI.).  Exactly  the  same  holds  for  B.  Here, 
too,  all  cross-cuts  can  be  so  drawn  that  they  likewise  form 
cross-cuts  in  T,  and  if  s  be  the  number  of  non-dividing  cross- 
cuts possible  in  B,  s  cannot  be  greater  than  q  and  is  therefore 
a  finite  number.  By  drawing  these  s  cross-cuts  B  is  made 
simply  connected.  Now  if  all  these  cross-cuts,  and  jR  also,  be 
drawn,  we  have  obtained  two  simply  connected  pieces  by  means 
of  r  +  s  +  1  cross-cuts.  Therefore,  according  to  Eiemann's 
fundamental  proposition, 

r  +  s  =  q. 

Of  these  q  cross-cuts,  r  run  entirely  in  A,  the  other  s  entirely 
in^. 

VI.  If  a  (q  +  lyply  connected  surface  T  he  resolved  by  means 
of  V  non-dividing  and  s  dividing  cross-cuts  (which  may  be  drawn 
in  any  order)  into  s-f-l  distinct  pieces  A^,  A^  A^,  •••,  A„  and 
if  the  orders  of  connection  of  these  pieces  be  determined  by  the 
numbers  of  cross-cuts  r^,  ri,  r^,  •••,  r„  respectively,  then 

g  =  V  -t-  ro  +  ri  -f  ^2  H h  n- 

Since  by  the  last  proposition  a  surface  of  finite  order  is 
always  resolved  by  a  dividing  cross-cut  into  two  pieces  which 
are  also  of  finite  orders,  these  orders  will  still  be  finite  if  a 
series  of  non-dividing  cross-cuts  be  drawn  in  T  before  the 
division.  The  same  conclusion  holds,  if  each  of  the  resulting 
pieces  be  further  resolved  in  like  manner.  Therefore  r^,  r^,  r^ 
•  ••,  r,  are  finite  numbers,  none  of  which  is  greater  than  q,  in 
whatever  order  we  may  have  drawn  the  v  -j-  s  cross-cuts.  Now 
if  all  the  pieces  A  be  further  changed  into  simply  connected 
surfaces  by  means  of  their  respective  r  cross-cuts,  we  have 
finally  s-\-\  simply  connected  pieces,  which  are  formed  by 
means  of  v  +  s  +  t-q  +  rj  +  rg  4-  •  •  •  -j-  r,  cross-cuts  in  all.  But 
we  likewise  obtain  s  + 1  simply  connected  pieces,  if  we  first 


SIMPLY  AND  MULTIPLY  CONNECTED   SUBFACES.      219 

make  T  simply  connected  by  means  of  q  cross-cuts  and  then 
by  means  of  s  additional  cross-cuts  resolve  it  into  s  -f  1  pieces. 
Therefore, 

g  +  s  =  V  +  s  +  ^'o  +  J'l  +  »'2  H f-  r„ 

or  q  =  v  +  n  +  ri  +  r.2-\ +  r,. 

VII.  If  a  (q  +  'i-)-ply  connected  surface  be  resolved  by  m 
cross-cuts  into  two  distinct  pieces,  one  of  which,  S,  is  simply 
connected,  then  the  other,  T',  is  (q  —  m  +  '^Yply  connected,  i.e., 
only  q  —(m  —  T)  cross-cuts  are  needed  to  modify  it  into  a  simply 
connected  surface. 

Let  X  be  the  number  of  cross-cuts  which  change  T'  into  a 
simply  connected  surface  T^'.  If  these  cuts  be  drawn,  we 
have  two  simply  connected  surfaces,  Tq  and  S,  formed  by 
means  of  m  +  a;  cross-cuts.  But  if  the  original  surface  be  first 
modified  into  one  simply  connected  by  means  of  the  appro- 
priate q  cross-cuts,  and  if  the  surface  so  formed  be  then  divided 
into  two  distinct  pieces  by  means  of  an  additional  cross-cut, 
we  have  again  formed  two  simply  connected  surfaces  by  means 
oi  q-\-l  cross-cuts.      Then,  by  the  fundamental  proposition 

(§  49), 

{m-{-x)-2=(q  +  l)-2, 

and  hence  x=  q  —(m  —  T). 

VIII.  If  a  surface  consisting  of  one  piece  possess  more  than 
one  boundary-line,  i.e.,  if  its  botindary  consist  of  several  distinct 
closed  lines,  it  is  multiply  connected. 

If  a  and  b  be  two  points  situated  on  different  boundary-lines, 
we  can,  since  the  surface  is  connected,  draw  a  line  from  a  to 
b  through  the  interior  of  the  surface.  This  is  a  cross-cut, 
which  does  not  divide  the  surface,  however,  for  we  can  come 
from  one  side  of  the  cross-cut  to  the  other  side  of  the  same  in 
the  surface  by  following  one  of  the  two  boundary-lines.  Since 
it  is  thus  possible  to  draw  in  the  surface  a  cross-cut  which 
does  not  divide  it,  the  surface  is  multiply  connected  (§  48, 1.). 

IX.  From  this  follows :  A  simply  connected  surface  always 
possesses  a  single  houndary-line,  i.e.,  its  boundary  can  be  traced 


220  THEORY  OF  FUNCTIONS. 

in  a  continuous  description.     (Or,  its  boundai'y  consists  of  only  a 
single  point.) 

Therefore,  after  a  multiply  connected  surface  has  been  modi- 
fied into  a  simply  connected  surface  by  means  of  cross-cuts, 
wherein  the  cross-cuts,  as  they  are  drawn,  are  to  be  added  to 
the  original  boundary  as  new  boundary-pieces,  it  must  be 
possible  to  trace  the  latter,  together  with  the  original  boundary, 
in  a  continuous  description.  Therein  each  cross-cut  simulta- 
neously forms  the  boundary  of  each  piece  of  the  surface  con- 
tiguous to  it  on  each  side.  If  then  the  entire  boundary  be 
traced  in  the  positive  direction,  so  that  the  bounded  region 
always  lies  on  the  left  of  the  boundary,  each  cross-cut  must 
be  traced  twice  in  opposite  directions.  (Cf.  Figs.  43  to  46, 
pp.  191  and  192.) 

X.  TTie  number  of  houndary-lines  is  either  increased  or  dimin- 
ished by  unity  by  every  cross-cut. 

According  to  the  discussion  of  §  47  a  cross-cut  always  fur- 
nishes two  boundary-edges,  because  it  simultaneously  bounds 
the  portions  of  the  surface  contiguous  to  it  on  each  side.  Now 
there  are  three  kinds  of  cross-cuts  (§  47) : 

(1)  The  cross-cut  may  join  two  points  a  and  b  of  the  same 
boundary-line.  The  latter  is  then  divided  into  two  parts  by 
the  points  a  and  b ;  one  part  forms  with  the  one  edge  of  the 
cross-cut  one  boundary-line,  the  other  part  forms  Avith  the 
other  edge  a  second  boundary-line.  Thus  two  boundary-lines 
are  formed  from  one,  and  the  number  of  boundary-lines  is 
increased  by  unity .^ 

(2)  The  cross-cut  may  join  two  points  which  lie  on  different 
boundary-lines.  Then  it  unites  these  into  a  single  one,  because 
its  two  edges  establish  the  connection.  Thus  from  two  boun- 
dary-lines is  formed  one,  and  the  number  of  boundary-lines  is 
diminished  by  unity. 

(3)  The  cross-cut  may  begin  at  a  point  of  the  boundary  and 

1  This  result  still  holds  if  the  poiuts  a  and  b  approach  each  other  and 
finally  coincide. 


SIMPLY  AND  MULTIPLY  CONNECTED   SURFACES.      221 

end  at  a  point  of  its  previous  course.  Then  one  of  its  edges, 
together  with  the  boundary-line  from  which  it  starts,  forms  a 
single  closed  boundary-line.  But  in  addition  the  inner  edge 
of  its  closed  portion  forms  a  new  boundary-line,  so  that  the 
number  of  boundary-lines  is  increased  by  unity. 

XI.  If  a  closed  surface  (which  therefore  j^ossesses  but  one 
boundary-point)  be  multiply  connected,  and  if  it  can  be  modified 
into  a  simply  connected  surface  by  means  of  a  finite  number  of 
cross-cuts,  the  number  of  such  cross-cuts  is  ahvays  even. 

Let  the  given  surface  be  (q  +  l)-ply  connected,  so  that  q 
cross-cuts  modify  it  into  a  simply  connected  surface.  Since 
the  surface  originally  possesses  only  a  single  boundary -point, 
the  number  of  its  boundary-lines  is  1.  This  number  is  either 
increased  or  diminished  by  unity  by  every  cross-cut  (X.).  Let 
p  be  the  number  of  cross-cuts  which  produce  an  increase,  and 
therefore  q  —  p  the  number  which  produce  a  diminution  in 
the  number  of  boundary -lines ;  then  at  the  end  the  number 
of  boundary-lines  is  '^  +  p  —{q—p}-  But  since  the  surface  is 
then  simply  connected,  it  possesses  only  one  boundary-line 
(IX.) ;  accordingly  we  have 

l+p-q+p  =  l, 

and  hence  q  =  2p. 

Thus  q  is  an  even  number. 

54.  If  we  know  the  number  of  sheets,  as  well  as  branch- 
points, in  a  surface  closed  at  infinity,  we  can  determine  its 
order  of  connection.  For  this  purpose,  as  in  §  13,  we  regard 
a  winding-point  of  the  (m  —  l)th  order  as  resulting  from  the 
coincidence  of  m  —  1  simple  branch-points.  If  in  this  sense 
g  be  the  number  of  simple  branch-points,  n  the  number  of 
sheets,  and  q  the  number  of  cross-cuts  which  modify  the  sur- 
face into  one  simply  connected,  we  can  find  a  relation  between 
these  three  numbers.^ 

1  Cf .  with  the  following:  Roch,  "  Ueber  Funktionen  complexer  Gros- 
sen,"  Schlomilch's  Zeitschr.  f.  Math.,  Bd.  10,  S.  177. 


222  THEORY  OF  FUNCTIONS. 

Let  A^  be  the  boundary-point  to  be  assumed  in  the  closed 
surface.  We  will  also  remove  from  the  surface  n  —  1  other 
points  Ax,  Ai,  •••,  A„^i,  one  from  each  of  the  n  —  1  other 
sheets ;  for  greater  simplicity  let  us  assume  that  these  ri  points 
lie  directly  below  one  another.  Now,  since  the  order  of  con- 
nection is  increased  by  unity  through  the  removal  of  every 
such  point  (§  53,  III.),  the  order  in  this  case  is  increased  by 
n  —  1.  Therefore  q  -{-  n  —  1  cross-cuts  are  necessary  to  modify 
the  surface  into  a  simply  connected  surface  after  the  removal 
of  the  n  —  1  points  Ai,  A^,  •••,  A„_-i.  In  this  (q  +  ^')'P^y  ^^^^' 
nected  surface  let  us  now  draw  cross-cuts  in  the  following  way. 
From  each  point  A  let  us  draw  lines  to  all  the  branch-points 
which  lie  in  the  same  sheet  with  A.  Then  it  is  evident  that 
thereby  we  have  actually  formed  cross-cuts,  because  we  can 
pass  through  a  branch-point  into  all  those  sheets  which  are 
connected  at  that  point.  If  two  points  A,,  and  A^  lie  in  two 
sheets  which  are  connected  at  a  simple  branch-point  a,  then 
A^a  and  aA^  together  form  a  line  which  leads  from  one  bound- 
ary-point A„  through  the  interior  of  the  surface  to  a  boundary- 
point  A^;  thus  A^aA/,  is  a  cross-cut.  On  the  other  hand,  if 
a  be  a  winding-point  of  the  (m  —  l)th  order,  at  which  are  con- 
nected m  sheets  containing  the  points  A^,  Ao,  •••,  A„,  say,  then 
first  A^aA.,  is  a  cross-cut,  but  after  it  is  drawn  each  one  of  the 
m  — 2  other  lines  aA^,  aA^,  •••,  aA„  becomes  a  cross-cut; 
therefore  in  this  case  we  have  in  all  m  —  1  cross-cuts,  just  as 
many  as  there  are  simple  branch-points  united  in  a.  If  we 
proceed  in  this  way  with  all  the  branch-points,  we  obtain 
exactly  as  many  cross-cuts  as  there  are  simple  branch-points, 
i.e.,  g.  But  these  g  cross-cuts  resolve  the  surface  into  n  distinct 
pieces,  each  by  itself  simply  connected ;  that  is,  the  ?i  sheets 
of  the  surface  are  separated  from  one  another  by  them  in  a 
certain  manner.  For,  if  p^  and  p^  be  two  points  lying  one 
above  the  other  in  any  two  sheets,  we  can  come  from  jh  to  jh 
only  by  crossing  branch-cuts  and  winding  round  branch-points ; 
but  the  latter  is  rendered  impossible  by  the  cross-cuts  con- 
structed. Thus  every  two  such  points  always  lie  in  distinct 
pieces.    Only  the  points  A  furnish  exceptions.    We  can  always 


SIMPLY  AND  MULTIPLY  CONNECTED   SURFACES.      223 

come  from  any  one  point  A  to  another  point  A  by  crossing  a 
branch-cut.  The  n  sheets  of  the  surface  are  therefore  sepa- 
rated from  one  another  in  this  way :  in  every  sheet  an  angular 
piece  (or  also  several  such  pieces),  which  is  formed  by  two 
cross-cuts  meeting  at  the  point  A,  is  separated  from  the  sheet, 
and  in  its  place  enters  a  corresponding  angular  piece  of  another 
sheet.  Accordingly  the  surface  consists  of  n  distinct  portions. 
But  each  of  these  is  by  itself  a  connected  piece ;  for,  since  it 
is  closed  at  infinity,  its  boundary  consists  solely  of  the  cross- 
cuts which  meet  in  the  point  A.  For  the  same  reason,  also, 
each  portion  is  by  itself  simply  connected,  since  we  can  come 
from  only  one  side  of  every  closed  line  drawn  in  it  to  that 
boundary;  thus  each  closed  line  forms  a  complete  boundary. 
Therefore,  after  the  removal  of  the  n  —  1  boundary-points  A^, 
A2,  ••',  A„_i,  the  given  surface  is  resolved  by  g  cross-cuts  into 
n  distinct  pieces,  each  by  itself  simply  connected.  But  this 
surface  was  (q  +  «)-ply  connected ;  therefore  g  -f  n  —  1  cross- 
cuts are  necessary  to  modify  it  into  a  simply  connected  surface. 
Now,  to  divide  the  latter  into  n  distinct  pieces,  n  —  1  additional 
cross-cuts  are  necessary  (§  48,  V.) ;  accordingly  the  original 
surface  is  divided  into  n  distinct  simply  connected  pieces  by 
means  of  g  +  2(«  —  1)  cross-cuts.  But  the  same  number  was 
found  above  equal  to  g,  and  therefore,  by  the  fundamental 
proposition  (§  49), 

g=g  +  2(n  -  1)  or  q  =  g-2(n-  1). 

Compare  with  this  result  the  examples  given  in  §  46  to 
which  it  is  applicable.  In  the  third,  n  =  2,  g  =  2;  therefore 
q  =  0,  and  the  surface  is  simply  connected.  In  the  fifth 
example  n  =  2,  g  =  4=',  therefore  q  =  2,  and  the  surface  is 
triply  connected. 

From  the  result  obtained  above  we  may  draw  certain  con- 
clusions. For,  since  in  a  closed  surface  q  is  always  an  even 
number  (§  53,  XI.),  g  must  also  be  even.  Therefore  a  surface 
closed  at  infinity  always  possesses  an  even  number  of  simple 
branch-points.  With  an  ?i-sheeted  surface  the  simplest  case 
would  be  the  occurrence  of  two  branch-points  of  the  (n  —  l)th 


224  THEORY  OF  FUNCTIONS. 

order;   and  if  this   be  the  case,  the  surface  is  simply  con- 
nected. 

A  further  inference,  which  follows  from  the  preceding,  is 
this,  that  a  surface,  which  serves  to  distribute  the  values  of 
an  algebraic  function  w  so  that  this  becomes  a  uniform  function 
of  position  in  the  surface  (§  12),  always  possesses  a  finite 
order  of  connection,  and  therefore  can  be  changed  by  a  finite 
number  of  cross-cuts  into  a  simply  connected  surface.  For 
the  number  of  sheets  n  is  equal  to  the  number  of  values  which 
the  function  lo  possesses  for  each  value  of  the  variable,  and  is 
therefore  a  finite  number.  That  the  number  g  of  simple  branch- 
points is  also  finite,  follows  from  the  fact  that  the  branch- 
points are  to  be  sought  for  only  among  those  points  at  which 
values  of  the  function  become  either  equal  or  infinite  (§  8). 
The  number  of  the  latter  points  is  finite  (§  38).  But  if 
F(w,  z)  =  0  denote  the  equation  of  the  /ith  degree  by  which  w 
is  defined,  the  points  z  at  which  values  of  the  function  become 
equal  are  those  which  simultaneously  satisfy  the  equations 

i^K.)=Oand^^^=0. 

By  the  elimination  of  w  from  these  equations  we  obtain  an 
equation  of  finite  degree  in  z.  Moreover,  since  at  most  n  values 
can  become  equal  at  each  of  these  points,  and  since  therefore 
at  each  branch-point  n  sheets  at  most  can  be  connected,  each 
branch-point  is  of  finite  order.  Accordingly  n  and  g  are  finite 
numbers,  and  hence  g  is  also  finite. 

55.  From  the  result  of  the  preceding  paragraph  we  can  also 
derive  a  relation  between  the  order  of  connection  of  an  unclosed 
surface  extended  in  a  plane,  the  number  of  its  simple  branch- 
points and  the  number  of  circuits  made  by  its  boundary. 

We  begin  with  a  surface  closed  at  infinity.  Let  this  be 
(g'-f  l)-ply  connected ;  let  gr'  be  the  number  of  its  simple  branch- 
points and  «  the  number  of  its  sheets.  Then,  according  to 
the  preceding  paragraph,  we  have 

g'  =  g'-2{n-l). 


SIMPLY  AND  MULTIPLY  CONNECTED   SURFACES.      225 

We  will  now  assume  that  the  boundary-point  which  is  to  be 
assigned  to  the  surface  lies  at  the  point  at  infinity  of  one 
of  its  sheets,  and  first  premise  that  in  no  sheet  is  the  point  at 
infinity  a  branch-point.  If  we  then  remove  from  each  sheet 
a  piece  which  contains  its  point  at  infinity,  and  which  is  there- 
fore bounded  by  a  line  returning  simply  into  itself,  the  order 
of  connection  of  the  surface  is  increased  by  unity  for  every 
piece  removed,  with  the  exception  of  that  which  contains  the 
assumed  boundary-point  (§  53,  IV.).  Thus  the  order  of  con- 
nection is  increased  by  n  —  1.  If,  therefore,  the  new  surface 
be  {q  +  l)-ply  connected,  we  have 

q  =  q'  +  n  —  l, 

and  consequently  q  =  g'  —  n  +  1. 

But  after  the  points  at  infinity  have  been  removed  from  the 
surface,  we  can  assiime  its  sheets,  which  were  previously  to 
be  conceived  as  infinitely  great  spherical  surfaces,  to  be  again 
extended  in  the  plane.  Each  sheet  then  appears  bounded  by 
a  line  returning  simply  into  itself,  which  makes  a  positive 
circuit  if  it  be  described  in  the  direction  of  increasing  angles. 
Consequently,  if  U  denote  the  number  of  circuits  forming  the 
boundary,  we  have  U=  n.  The  number  of  simple  branch- 
points g  contained  in  the  new  surface  is  equal  to  the  previous 
number  g',  since  by  the  assumption  no  branch-point  was  re- 
moved from  the  surface.  We  therefore  obtain  from  the  last 
equation 

q  =  g-U-{-l. 

This  is  the  relation  mentioned  above,  and  it  will  now  be  shown 
that  it  does  not  lose  its  validity  when  certain  changes  are  made 
in  the  surface. 

Let  us  first  consider  the  case  when  m  sheets  in  the  original 
surface  are  connected  at  a  point  at  infinity,  when  therefore 
m  —  1  simple  branch-points  are  united  in  that  point.  Then 
the  number  of  pieces  removed  is  no  longer  equal  to  w  as  before, 
but  since  one  of  them  is  bounded  by  a  line  which  makes  m 
circuits  round  a  branch-point,  and  since  it  thus  takes  the  place 


226  THEOltY  OF  FUNCTIONS. 

of  m  of  the  previous  pieces,  that  number  is  equal  to  only 
n  —  m  +  l.  Moreover,  that  piece  which  contains  the  assumed 
boundary-point  does  not  cause  any  increase  in  the  order  of 
connection ;  accordingly  that  increase  amounts  to  n  —  m,  or 

q  =  q'  +  n  —  m, 

that  is,  Q^g'  —  2(n  —  1)  +  n  —  m 

=  g'  —  m  +  l  —  n-\-l. 

When  the  sheets  are  extended  in  the  plane  the  number  of  cir- 
cuits U  is  again  equal  to  n;  for  the  only  change  in  this  respect 
is  that  all  the  n  boundary-lines  are  no  longer  distinct,  but  7n 
of  them  are  united  into  a  single  one,  which  now,  however, 
makes  m  circuits.  On  the  other  hand,  m  —  1  simple  branch- 
points are  removed  from  the  surface  with  the  points  at  infinity ; 
therefore  now 

g  —  g'  —  m  +  l. 

If  we  substitute  this  value  of  g'  in  the  last  equation,  we  obtain 
again  as  before 

q  =  g-U+l. 

We  will  now  modify  the  w-sheeted  surface  extended  in  the 
plane  by  removing  places  in  the  interior. 

Let  us  first  consider  a  closed  line  bounding  a  portion  of  the 
surface  which  does  not  contain  a  branch-point,  and  let  us 
imagine  this  piece  removed.  Then,  in  the  first  place,  q  is  in- 
creased by  + 1  (§  53,  IV.).  But  the  new  boundary-line,  if 
its  boundary-direction  is  to  be  positive,  must  be  described  in 
the  direction  of  decreasing  angles.  Therefore,  if  we  now 
understand  in  general  by  the  number  of  positive  circuits  the 
positive  or  negative  number  U,  which  results  from  subtracting 
the  number  of  circuits  in  the  direction  of  decreasing  angles 
from  the  number  of  circuits  in  the  direction  of  increasing 
angles,  this  number  JJ  in  the  preceding  case  must  be  increased 
by  —  1.  At  the  same  time  q  is  increased  by  +  1,  and  thus 
the  above  relation  remains  unchanged. 

For  instance,  if  the  surface  consist  of  one  sheet,  then  g  =  0; 
if,  further,  it  be  bounded  by  one  outer  line  and  k  smaller 


SIMPLY  AND  MULTIPLY  CONNECTED   SURFACES.      227 

circles  enclosed  by  the  former,  then  the  outer  line  makes  a 
circuit  in  the  direction  of  increasing  angles  for  a  positive 
boundary-direction,  but  each  of  the  inner  circles  a  circuit  ia 
the  opposite  direction;  therefore 

U=l-  k, 
and  we  obtain  q  =  k  —  l  +  l  =  k; 

thus  the  number  of  cross-cuts  is  equal  to  the  number  of  inner 
circles. 

Secondly,  if  a  piece  of  the  surface  be  removed  which  con- 
tains a  branch-point  of  the  {m  —  l)th  order,  the  boundary-line 
of  which  therefore  makes  m  circuits,  g  is  again  increased  by 
-f  1  (§  53,  IV.),  U  at  the  same  time  by  —  m,  g  by  —  (m  —  1), 
and  therefore  g  —  U  hy  +1;  consequently  the  above  relation 
again  remains  the  same. 

After  the  modifications  introduced  hitherto  the  surface  con- 
sidered has  a  configuration  such  that  the  outer  boundary-lines 
enclose  all  branch-points  situated  in  the  finite  part  of  the 
surface;  it  also  has  gaps  in  the  interior,  yet  of  such  a  kind 
that  each  of  the  pieces  of  surface  removed  contains  either  no 
branch-point  or  only  one  (of  any  order).  We  have  now  to 
inquire  whether  the  above  relation  changes,  either  when  the 
outer  boundary-lines  no  longer  enclose  all  finite  branch-points, 
or  when  the  inner  boundary -lines  enclose  portions  of  the  sur- 
face that  were  removed  in  which  more  than  one  branch-point 
was  contained.  Both  conditions  lead  to  the  same  inquiry; 
namely,  to  the  examination  of  the  case  when  there  is  removed 
a  portion  of  the  surface  contiguous  to  an  (outer  or  inner)  edge 
which  contains  a  branch-point  of  the  (m  —  l)th  order,  but  no 
gaps.  The  latter  assumption  can  be  made  without  loss  of 
generality,  since  the  occurrence  of  gaps  has  already  been  dis- 
posed of  by  the  preceding  considerations.  Now  if  such  a  piece 
of  the  surface  is  to  be  removed,  then,  since  it  is  contiguous  to 
the  edge,  its  removal  must  be  effected  by  means  of  cross-cuts, 
and  these  must  be  dra^vn  in  such  a  way  that  the  boundary  of 
the  piece  removed,  consisting  of  the  cross-cuts  and  the  con- 
tiguous  parts    of   the    boundary,   forms   a  closed   line  which 


228  THEORY  OF  FUNCTIONS. 

makes  m  circuits  round  the  winding-point.  If  none  of  these 
cross-cuts  wind  round  the  branch-point,  then  m  cross-cuts  are 
necessary  to  that  end;  otherwise  a  less  number.  Since,  how- 
ever, the  piece  removed  is  bounded  by  a  single  actually  closed 
line,  it  is  simply  connected  (§  46,  Ex.  2).  We  will  now 
examine  the  case  in  which  no  cross-cut  winds  round  the  branch- 
point ;  then  a  simply  connected  piece  of  the  surface  is  removed 
by  means  of  m  cross-cuts,  and  consequently  the  order  of  con- 
nection of  the  surface  which  remains  is  diminished  by  m  —  1 
(§  53,  VII.).  At  the  same  time  g  is  diminished  by  m  —  1 
through  the  removal  of  the  winding-point  of  the  (m  —  l)th 
order.  But  the  number  U  suffers  no  change.  For,  since  the 
m  cross-cuts  add  no  new  positive  or  negative  circuits,  merely 
a  different  kind  of  connection  of  the  bomidary-line  is  produced 
by  the  removal  of  the  branch-point,  while  the  circuits  of  the 
same  remain  unchanged.  Thus,  since  q  and  g  are  each  dimin- 
ished by  m  —  1  and  U  remains  unchanged,  the  above  relation 
still  holds. 

Finally,  let  us  consider  the  case  in  which  the  boundary  is 
changed  by  means  of  cross-cuts  which  do  not  divide  the  sur- 
face. In  this  we  turn  our  attention  to  the  change  of  direction 
which  the  lines  experience,  and  remark  that  a  line  makes  a 
positive  circuit  if  it  experience  a  total  change  of  direction 
equal  to  2  tt.  If  a  non-dividing  cross-cut  be  drawn  in  the  sur- 
face, this  at  the  same  time  furnishes  two  boundary -pieces  and 
must  be  described  twice  in  opposite  senses  in  conforming  to 
the  positive  boundary-direction.  Where  the  cross-cut  meets 
a  part  of  the  boundary  of  the  surface, 
the  boundary-direction  experiences  an 
abrupt  change.  Let  a  be  the  angle  by 
which  the  direction  changes  (Fig.  50). 
(The  case  can,  it  is  true,  occur  in  which 
the  cross-cut  changes  into  a  part  of  the 
boundary  without  an  abrupt  change  of 
direction,  but  this  is  included  in  the  preceding  if  we  assume 
a  =  0.)  In  the  description  of  the  boundary-lines,  of  which 
the  cross-cut  always  forms  a  part,  we  once  more  return  to  the 


SIMPLY  AND  MULTIPLY  CONNECTED   SURFACES.      229 

former  place,  since  the  cross-cut  must  be  described  twice; 
then  the  cross-cut  is  described  in  the  opposite  direction,  but 
the  contiguous  part  of  the  original  boundary  in  the  same 
direction  as  before.  From  this  it  follows  that  the  boundary- 
direction  now  experiences  an  abrupt  change  equal  to  tt  —  «. 
Consequently  the  end-point  of  the  cross-cut  produces  a  total 
change  of  direction  equal  to  tt.  (Also  in  the  case  when  this 
ends  in  a  point  of  its  previous  course,  because  then  the  cross- 
cut itself  takes  the  place  of  the  original  boundary-line.)  The 
same  change  occurs  at  the  other  end  of  the  cross-cut.  There- 
fore the  cross-cut  causes  at  its  end-points  a  change  of  direction 
equal  to  2  tt.  On  the  other  hand,  the  change  of  direction 
experienced  by  the  cross-cut  during  its  course  need  not  be 
considered,  because  this  change  is  neutralized  by  that  of  the 
subsequent  description  in  the  opposite  sense.  Consequently 
each  non-dividing  cross-cut  increases  the  number  of  positive 
circuits  f7  by  -f  1 ;  ^  at  the  same  time,  however,  it  diminishes 
the  order  of  connection  by  unity  (§  53,  I.)  and  therefore  q  is 
increased  by  —  1.  Consequently  the  above  relation  holds  in 
this  case  also. 

According  to  the  preceding  considerations,  the  equation 
q  —  g  —  U+1  holds  for  all  surfaces  which  can  be  formed  by 

1  The  same  conclusion  holds  when  a  cross-cut  divides  the  surface  into 
two  distinct  pieces.  For,  since  a  cross-cut  which  joins  two  different 
boundary -lines  never  divides  the  surface  (§  63,  VIII.),  a  dividing  cross- 
cut can  either  merely  join  two  points  of  the  same  boundary-line,  or, 
starting  from  one  boundary-line,  end  in  a  point  of  its  previous  course. 
In  both  cases  two  boundary-lines  are  produced  by  it  from  one.  (§  53,  X. 
(1)  and  (3).  See  also  Fig.  40  and  Fig.  41,  p.  190.)  If  these  be  traced 
in  succession  in  the  positive  boundary-direction,  the  original  boundary- 
line  is  described  once,  but  the  cross-cut  twice  in  opposite  directions. 
Consequently  the  above  considerations  still  hold.  Therefore  if  f/"  be  the 
number  of  circuits  of  the  original  boundary-lines,  Ui  and  U2  the  numbers 
for  the  two  boundary-lines  resulting  from  the  cross-cut,  we  have 

Ui+  U2  =  U+\. 
It  is  evident  at  once  that  this  formula  does  not  lose  its  validity,  if  the  two 
boundary -lines  resulting  from  the  cross-cut  have  only  one  point  in  com- 
mon, in  which  case  two  pieces  of  the  original  boundary-line  approach 
each  other  and  the  cross-cut  drawn  at  this  place  is  infinitesimal  in  length. 


230'  THEORY  OF  FUNCTIONS. 

means  of  the  cross-cuts  discussed.  For  the  formation  of  many 
kinds  of  surfaces  (as,  for  instance,  those  represented  in  Rie- 
mann's  dissertation:  "Lehrsatze  aus  der  Analysis  situs," 
u.  s.  w.,  Crelle's  Joiim.,  Bd.  54,  S.  110,  last  example),  it 
would  be  necessary  still  to  consider  the  case  in  which  a  por- 
tion of  the  surface  contiguous  to  an  edge  and  containing  a 
branch-point  is  to  be  removed  by  means  of  cross-cuts  which 
wind  round  the  branch-point ;  in  another  place  ^  it  was  shown 
that  in  this  case  also  the  above  relation  does  not  lose  its 
validity.  But  we  cannot  always  affirm  with  certainty  that 
every  surface,  however  bounded,  could  be  produced  by  means 
of  such  cross-cuts,  as  long  as  we  do  not  know  in  advance  the 
form  of  a  particular  given  surface.  Therefore  we  will  in  pref- 
erence add  another  proof  for  the  general  validity  of  the  above 
relation. 

This  is  based  upon  the  property,  that  the  boundary-line  of 
a  simply  connected  surface,  extended  in  the  plane  and  con- 
taining no  branch-points, 
always  makes  but  one  cir- 
cuit, and  that  this  also  holds 
when  the  surface  has  first 
been  reduced  to  this  simple 
connection  by  means  of 
cross-cuts.  For  in  the  first 
place  only  one  boundary -line 
^"'-  5^-  can    ever    occur   in   such  a 

surface  (§  53,  IX.).  But  if  we  represent  this  as  a  movable 
thread,  we  can  show  that  it  can  always  be  deformed  into  a 
circle  which  is  to  be  described  once.  For,  since  the  boundary- 
line  nowhere  intersects  itself,  and  also  since  its  displacement 
is  nowhere  prevented  by  a  branch-point,  the  deformation  into 
a  circle  could  be  made  impossible  only  by  the  line  somewhere 
forming  a  loop  which  could  not  be  opened  by  enlarging.  But 
if  this  be  the  case,  the  portions  of  the  surface  which  are  con- 
tiguous to  the  boundary-line  where  this  forms  the  loop,  and 

1  "Zur  Analysis  situs  Riemann'scher  Flachen,"  Ber.  d.  Wien.  Akad., 
Bd.  69,  Abth.  II.,  Januar  1874.     See  here  Fig.  1. 


SIMPLY  AND  MULTIPLY  CONNECTED  SURFACES.      231 

which  thus  pass  one  over  the  other,  must  later  be  connected 
with  each  other  in  their  continuations  beyond  A  and  B 
(Fig.  51).  If  these  portions  always  remain  separated  beyond 
A  and  B,  the  loop  can  be  at  once  opened  by  enlarging.  But  if 
A  and  B  be  connected,  we  can,  by  drawing  a  cross-cut  from  a 
point  of  the  loop,  come  from  the  one  side  of  the  same  to  the 
other  beyond  A  and  B,  since  these  are  connected;  thus  the 
cross-cut  does  not  divide  the  surface  and  this  is  multiply  con- 
nected (§  48,  I.).  Accordingly  every  loop  which  occurs  in  a 
simply  connected  surface  can  always  be  opened,  and  the 
boundary-line  therefore  be  deformed  into  a  circle.  If  the 
latter  be  described  in  the  direction  of  increasing  angles,  it 
forms  a  single  positive  circuit. 

Assume  now  an  arbitrary  Riemann  surface  extended  over 
a  finite  part  of  the  plane,  and  let  q,  g,  U  have  for  this  their 
former  meanings.  Then  g  —  U-\-l  is  always  an  integer  (or 
zero).  The  formula  to  be  proved  asserts  that  this  number  is 
exactly  equal  to  q.  We  will  now  not  presuppose  this,  but  will 
assume 

g-U+l^q  +  Tc, 

and  then  prove  that  k  must  be  zero.  To  this  end  we  first 
remove  all  branch-points  from  the  surface,  by  enclosing  each 
one  in  an  actually  closed  line  and  removing  from  the  surface 
the  piece  so  bounded  which  contains  the  branch-point.  Then  the 
last  equation  still  holds ;  for,  as  was  previously  shown,  p.  227, 
for  the  removal  of  a  branch-point  of  the  (m  —  l)th  order,  q 
changes  into  g  -f- 1,  gr  into  g  —  m-\-l,  ?7into  U—m,  and  hence 
g  —  U  into  g  —  U-{-l.  Therefore,  if  q  change  into  q',  U  into 
If  after  the  removal  of  all  the  branch-points,  we  obtain,  since 
g  becomes  zero, 

_  C7' +  1  =  g' +  A:. 

For  the  modification  of  the  surface  into  one  simply  connected, 
g'  non-dividing  cross-cuts  are  requisite.  If  these  be  drawn,  q' 
changes  into  g'  —  1  for  each  one,  and,  by  p.  229,  IT  at  the  same 
time  into  IT  -\-l.  Consequently  the  preceding  equation  still 
holds.     Hence  if  If  change  into  XT'  when  the  surface  becomes 


i232  THEOBY  OF  FUNCTIONS. 

simply  connected,  and  when  therefore  q'  becomes  zero,  we 
have 

But  now  this  surface  is  not  only  simply  connected,  but  it  no 
longer  contains  a  branch-point;  its  boundary -line  therefore 
makes  but  one  circuit,  i.e.,  U"  =  -\-l,  and  consequently  A;  =  0; 
which  was  to  be  proved. 

For  a  simply  connected  surface  (g  =  0)  the  equation 

q=g-U+l 

changes  into  JJ=  g  +  1. 

Accordingly  the  proposition,  which  we  found  to  be  valid  for 
a  special  case,  §  13,  holds  generally :  The  number  of  circuits 
of  the  boundary-line  of  a  simply  connected  surface  is  greater  by 
unity  than  the  number  of  simple  branch-points  in  its  interior. 
Yet  it  is  well  to  notice  that  the  validity  of  this  relation,  as 
well  as  of  the  more  general  one  q  =  g  —  U+1,  depends  upon 
the  surface  being  extended  in  the  plane. 

56.  We  will  now  also  examine  such  Riemann  surfaces  as 
cannot  be  extended  in  a  plane,  inquire  under  what  conditions 
their  orders  of  connection  are  finite,  and  determine  these  orders 
more  exactly.  At  the  same  time  we  first  premise  that  the  sur- 
face possesses  only  a  finite  number  of  sheets  and  branch-points, 
and  assume  that  none  of  its  boundary-lines  pass  through  a 
branch-point. 

We  begin  with  a  closed  surface,  which  therefore  possesses 
only  one  boundary-point  a.  We  will  call  this  a  complete  surface 
and  designate  it  by  W.     By  §  54,  the  relation 

q  =  G-'2{n- 1)  (1) 

holds  for  this  surface,  if  G  denote  the  number  of  simple 
branch-points  contained  in  it,  n  the  number  of  its  sheets,  and 
Q  the  number  of  its  cross-cuts.  Accordingly,  Q  is  a  finite 
number,  if  G  and  n  be  finite.  '  Hence  our  further  investigations 
relate  exclusively  to  the  boundary-lines. 

The  boundary-lines  which  are  to  be  introduced  into  a  com- 
plete surface  must  be  furnished  by  cuts  drawn  in  the  surface. 


SIMPLY  AND  MULTIPLY  CONNECTED   SURFACES.     233 

Either  these  do  not  divide  the  complete  surface,  or  they  divide 
it  and  remove  from  it  single  surface-pieces.  Accordingly  we 
distinguish  two  kinds  of  boundary -lines. 

By  boundary-lines  of  the  first  kind  we  understand  such  as  do 
not  remove  a  piece  from  a  complete  surface.  They  are  there- 
fore characterized  by  the  condition  that  in  the  new  surface  T 
two  edges,  which  belong  either  to  different  boundary-lines  or 
to  one  and  the  same  boundary-line,  run  everywhere  infinitely 
near  each  other.  If  we  consider  only  the  lines  along  which 
the  cuts  are  made,  without  regarding  the  edges  furnished  by 
them,  these  lines  form  a  line-system,  and  the  portions  of  the 
surface  T  itself  which  are  contiguous  to  the  two  sides  of  each 
line  belong  with  that  line. 

Secondly,  it  may  happen  that,  when  pieces  are  removed  from 
W  by  the  cuts  introduced,  so  that  gaps  occur  in  T,  the  edges 
belonging  to  the  boundary-lines  which  are  furnished  by  the 
cuts,  likewise  run  for  considerable  distances  infinitely  near 
one  another  in  isolated  places.  We  will,  however,  lay  partic- 
ular stress  on  those  boundary-lines  in  connection  with  which 
this  is  not  the  case,  and  call  them  by  the  distinctive  name 
hoxmdary-Unes  of  the  second  kind.  Consequently  a  boundary- 
line  of  the  second  kind  is  formed  by  a  closed  line,  and  every- 
where on  one  side  of  this  line  there  borders  a  portion  of  the 
surface  belonging  to  T,  while  on  the  other  side  a  gap  occurs. 
Thus  in  connection  with  a  boundary-line  of  the  second  kind 
two  edges  never  run  infinitely  near  each  other  for  any  dis- 
tance ;  ^  on  the  contrary,  when  in  connection  with  boundary- 
lines  which  divide  the  surface  two  edges  do  so  run,  a  boun- 
dary-line of  the  first  kind  is  connected  with  a  boundary-line 
of  the  second  kind.  If  the  boundary -lines  of  the  second  kind 
be  conceived  in  this  way,  each  boundary-line  is  either  of  the 
first  or  of  the  second  kind,  or  it  is  a  combination  of  the  two. 

We  will  now  show  that  every  surface  T,  which  possesses 
arbitrary  boundary-lines,  can  be  obtained  from  an  appropriate 

1  The  case  in  which  edges,  whicli  belong  to  boundary-lines  of  the  second 
kind,  come  infinitely  near  one  another  at  isolated  points  will  be  considered 
later. 


234  THEORY  OF  FUNCTIONS. 

complete  surface  by  means  of  cuts.  Let  us  first  assume  that 
T  contains  only  boundary -lines  of  the  second  kind.  In  this  case 
it  can  first  be  made  into  a  complete  surface  W  by  the  addition 
of  surface-pieces  B,  and  then  be  obtained  again  from  this  sur- 
face by  means  of  cuts.  This  is  at  once  evident,  if  each  one  of 
the  boundary-lines  which  enclose  the  gaps  run  entirely  in  one 
and  the  same  sheet.  If,  on  the  other  hand,  a  boundary -line  run 
in  several  sheets,  we  assume  that  the  supplementary  piece  B 
consists  of  the  same  sheets,  by  imagining  each  sheet  extended 
beyond  the  edge.  In  a  place  where  the  boundary-line  passes 
from  one  sheet  into  another,  a  branch-cut  must  occur,  or  be 
capable  of  being  assumed,  in  T.  At  such  a  place  we  extend  the 
branch-cut  into  B,  let  it  end  in  5  at  a  branch-point,  and  assume 
that  the  connection  of  the  sheets  at  this  point  in  B  is  just  as  it 
actually  occurs  in  T.  This  can  be  effected  in  every  place  where 
it  is  necessary  independently  of  every  other  place,  and  depends 
only  upon  the  particular  connection  of  the  sheets  in  T  for  each 
branch-cut.  If  a  gap  be  bounded  by  several  boundary-lines, 
the  same  method  of  procedure  is  followed  for  each.  We 
thereby  obtain  a  surface  B  everywhere  contiguous  to  the 
boundary-lines,  which  contains  no  gaps,  and  which  is  also 
completely  bounded  by  these  boundary-lines,  since  the  latter 
bound  completely  the  gaps. 

If  any  surface  T,  which  possesses  arbitrary  boundary-lines, 
be  under  discussion,  we  imagine  all  boundary-lines  of  the  first 
kind  removed,  by  regarding  the  lines  along  which  they  run  as 
not  drawn.  We  then  supplement  the  surface,  in  the  manner 
outlined  above,  into  a  complete  surface  and  cut  from  this  first 
the  boundary -lines  of  the  second  kind.  This  done,  it  is  at  once 
evident  that  the  boundary-lines  of  the  first  kind,  whether  they 
occur  alone  or  in  connection  with  those  of  the  second  kind,  can 
be  cut  in  the  surface. 

Thus  we  can  always  regard  a  given  surface  T  as  one  formed 
from  a  complete  surface  Why  means  of  cuts.  Let  the  number 
of  surface-pieces  B  removed  from  W  by  boundary-lines  of  the 
second  kind  be  s.  All  cuts  made  in  W  run  along  certain  lines. 
We  will  now  suppose  these  lines  to  be  drawn  in  W;  then  they 


SIMPLY  AND  MULTIPLY  CONNECTED  SURFACES.     235 

all  together  form  a  line-system.  It  is  not  necessary  for  all  the 
lines  of  this  system  to  be  connected.  We  will  assume  that  it 
consists  of  r  distinct  systems  Li,  L^,  •••,  L^.  Each  L  forms  a 
connected  line-system  complete  in  itself,  but  it  may  at  the 
same  time  contain  several  boundary-lines.  Let  us  assume 
the  boundary-point  a,  which  is  to  be  assigned  to  W,  on  one  of 
the  systems  L,  and  let  a  boundary-point  be  also  taken  on  each 
of  the  r  —  1  other  systems.  Then  all  the  systems  are  exactly 
alike  in  this  respect,  —  that  each  one  is  connected  with  a 
boundary-point.  We  therefore  need  to  examine  more  closely 
only  one  of  these  systems ;  we  will  designate  it  indefinitely  by 
L,  and  the  boundary-point  on  it  by  a. 

Since  L  is  by  itself  wholly  connected,  and  hence,  with  an 
exception  to  be  mentioned  immediately,  contains  no  simply 
closed  lines,  it  can  be  decomposed  into  simple  line-segments 
(§  50).  (The  exception  referred  to  occurs  when  L  consists  of 
a  single  simply  closed  line,  which  therefore  begins  at  a  point  a 
and  also  ends  at  that  point ;  but  in  that  case  L  forms  one  cross- 
cut.) If  we  denote,  as  in  §  50,  the  numbers  of  end-  and  nodal- 
points  contained  in  L  by  e,  k^,  k^,  kg,  •••,  then,  by  §  50,  L 
consists  of 

^(e  +  k,  +  2  ki+ 3  k, -}-•.•) 

simple  line-segments ;  or  of 

simple  line-segments,  if  for  brevity  we  let 

ks  +  2ki  +  3k,-] =  K 

A  simple  line-segment  is  a  cross-cut,  if  both  of  its  end-points 
lie  on  an  edge ;  but  if  one  end-point  lie  in  the  interior  of  the 
surface,  the  simple  line-segment  is  a  slit  (§  53,  II.).  Hence 
the  system  L  in  general  consists  partly  of  cross-cuts,  partly  of 
slits.  But  the  latter  need  not  be  considered,  because  a  slit, 
which  begins  at  the  edge  and  ends  in  the  interior,  can  never 
divide  the  surface  and  does  not  change  the  order  of  connection 
(§  53,  II.) ;  its  effect,  on  the  contrary,  consists  only  in  extend- 
ing a  boundary-line  which  already  exists.     Hence  it  is  impor- 


236  THEORY  OF  FUNCTIONS. 

tant  to  determine  the  number  p  of  cross-cuts  contained  in  L. 
To  that  end  we  remark  that  the  line-system  L,  according  to 
Lippich's  proposition,  can  in  fact  be  regarded  as  a  system  of 
cross-cuts,  but  only  when  conditions  (1)  and  (2)  of  §  51  are 
fulfilled.  This  is  not  the  case  if  the  end-points  of  L  lie  in  the 
interior  of  the  surface.  We  can,  however,  remove  these  by 
cutting  off  every  line-segment  which  contains  such  a  point  at 
the  nodal-point  nearest  to  it.  Then  the  system  which  remains, 
since  it  satisfies  conditions  (1)  and  (2),  consists  of  only  a  definite 
number  p  of  cross-cuts ;  but  the  line-segments  cut  off  become 
slits.  We  will  now  always  place  the  boundary-point  a,  which 
can  indeed  be  arbitrarily  assumed,  either  at  an  end-point  or 
at  an  ordinary  point.  If  a  be  an  end-point,  only  e  —  1  line- 
segments  are  to  be  cut  off,  since  a,  as  a  boundary-point,  need 
not  be  removed.  To  find  p,  therefore,  we  must  deduct  from 
the  number  ^(e  -|-  K)  of  all  the  simple  line-segments  the 
number  e  —  1  of  segments  which  are  not  cross-cuts,  and  we 
thereby  obtain 

p  =  ^(e^K)-(e-l)=^(K-e)  +  l. 

We  obtain  the  same  value  if  a  be  an  ordinary  point.  Then  all 
the  e  end-points  must  be  removed ;  but  now,  in  order  to  decom- 
pose the  system  which  remains  into  cross-cuts,  since  all  the 
line-segments  which  end  at  a  must  be  regarded  as  possessing 
end-points  at  that  place,  we  must,  according  to  the  discussion 
of  §  51,  count  a  as  two  end-points  in  addition.  Thus  the  whole 
number  of  simple  line-segments  is  now 

^(2  +  e  +  K), 
and  therefore 

p  =  x(2  +  e  +  iq-e  =  ^(K-e)  +  l, 

as  before;  and  this  relation  is  also  valid  for  the  exceptional 
case  mentioned  above. 

Consequently  each  of  the  r  line-systems  L  contains 


SIMPLY  AND  MULTIPLY  CONNECTED   SURFACES.     237 
cross-cuts,  and  therefore  they  all  in  the  aggregate  contain 

cross-cuts. 

If  we  now  refer  the  letters  e,  kg,  fc^,  ^5,  •••,  and  K  to  the  end- 
and  nodal-poiuts  of  the  entire  line-system  formed  by  all  the 
boundary -lines ;  that  is,  if  we  write  e  and  K  instead  of  2e  and 
2/ii  we  obtain  for  the  number  p  of  cross-cuts  which  are  formed 
by  all  the  boundary -lines  the  value 

i)  =  igi-e)-f  r.  (2) 

Some  of  these  cross-cuts  divide  the  surface;  others  do  not 
divide  it.  According  to  the  assumption  s  surface-pieces  B 
were  removed  from  W;  thus,  including  the  piece  T  which 
remains,  W  is  divided  into  s  -f  1  pieces.  Hence  s  is  the  num- 
ber of  dividing  cross-cuts,  because  no  cross-cut  can  divide  a 
surface  into  more  than  two  pieces,  and  none  can  cross  another. 
(Cf.  §  48,  V.)  If,  moreover,  we  denote  by  v  the  number  of 
non-dividing  cross-cuts,  we  have 

v-fs  =  |(/f-e)+r.  (3) 

Now  by  (1)  the  relation 

Q=G-2{n-l)  (4) 

held  for  the  surface  W;  but  r  —  1  boundary-points  were  re- 
moved from  W,  and  therefore  Q  must  be  increased  by  r  —  1. 
This  (Q  +  *')-ply  connected  surface  has  now  been  divided  by 
V  +  s  cross-cuts  into  s  -\-l  pieces ;  accordingly  the  proposition 
proved  in  §  53,  VI.  can  be  applied.  If  we  denote  by  q  the 
number  of  cross-cuts  for  T,  and  by  g/,  qj,  •••,  q,'  the  numbers 
for  the  s  surface-pieces  B  which  were  removed,  then  by  VI., 
§  53,  we  have 

Q  +  r-l  =  v  +  q  +  lqJ.  (5) 

From  this  equation  and  (3)  we  obtain 


238  THEORY  OF  FUNCTIONS. 

We  can  find  from  this  formula  when  g  is  a  finite  number. 
For,  since  Q,  and  by  VI.,  §  53,  also  every  q^',  is  finite,  q  remains 
finite  if  s,  K,  e  be  finite,  i.e.,  if  the  boimdary-lines  form  a  finite 
line-system  (in  the  sense  of  §  50). 

If  T  possess  only  boundary-lines  of  the  first  kind,  we  can 
quite  generally  determine  the  number  of  its  cross-cuts.  For 
in  this  case  the  cross-cuts  contained  in  the  boundary-lines 
are  all  non-dividing,  and  therefore  Q  is  first  to  be  increased  by 
r  —  1,  and  then  to  be  diminished  by  the  number  p  of  cross-cuts 
which  are  contained  in  the  boundary-lines.  But  the  slits, 
which  can,  moreover,  only  enter  as  boundary -lines  of  the  first 
kind,  or  as  parts  of  such  lines,  do  not  change  the  order  of  the 
surface.     We  therefore  have 

q=Q-\-r-l  —p, 

or  by  (4)  q=  G  —  2n  +  l  +  r  —  p', 

and  finally  by  (2)     q  =  g -2n  +  l  -  \{K- e),  (6) 

if  at  the  same  time  the  number  of  simple  branch-points  con- 
tained in  T,  which  in  this  case  is  equal  to  G,  be  again  denoted 

But  if  T  also  possess  boundary-lines  of  the  second  kind,  we 
will,  in  order  to  obtain  a  definite  expression  for  q,  make  a 
limiting  hypothesis ;  namely,  that  all  the  surface-pieces  B 
which  are  removed  can  be  extended  in  planes.^ 

For  this  surface  the  relation 

q^g-U+1 

of  §  55  can  be  applied.  If  we  denote  the  numbers  of  simple 
branch-points  contained  in  the  surface-pieces  B  by  g^,  g.,',  •••, 
g,',  we  have 

G  =  g  +  ^j,'.  (7) 

1  If  a  complete  surface,  say  a  Riemann  many-sheeted  spherical  sur- 
face, be  resolved  by  any  cuts  whatever  Into  two  distinct  pieces,  it  is  quite 
evident  that  the  cases  may  occur  in  which  either  both  pieces  or  only  one 
of  the  two  can  be  extended  in  a  plane  ;  but  it  is  very  probable  that  the 
third  case  may  also  occur  in  which  neither  can  be  extended.  In  the 
latter  case  the  following  investigation  would  lose  its  validity. 


*v 


SIMPLY  AND  MULTIPLY  CONNECTED  SURFACES.     239     ^''^    ^ 

If,  further,  U^  be  the  number  of  circuits  of  the  boundary-line^   ^^. 
in  one  of  the  pieces  B,  then  for  this  piece  Jjy^  ^ 

Therefore,  if  we  let  V='S,U^, 

we  obtain  for  the  aggregate  of  surface-pieces  B 

If  we  subtract  this  equation  from  (4)  and  attend  to  (7),  we 
obtain 

Q-:S.q,'  =  g-2n  +  2  +  V-s; 

and  since  from  (5) 

it  follows  that 

q  +  v  —  r  +  1  =9'  — 2/1  +  2  +  F—  s 

or  q  =  g  —  271  +  1— (v  +  s  —  r)+V, 

and  finally  by  (3) 

q  =  g-2n-{-l-^(K-e)-\-V.  (8) 

If  there  be  no  boundary-lines  of  the  second  kind,  and  hence 
if  1^=  0,  this  formula  reduces  to  (6). 

The  circuits  V  of  the  boundary-lines  of  the  second  kind  are, 
according  to  the  preceding,  to  be  counted  in  the  pieces  B  which 
are  removed,  and  in  the  way  specified  in  §  55,  namely :  Each 
boundary-line  is  to  be  so  described  that  the  piece  B  lies  on  the 
left ;  and  after  B  is  extended  in  a  plane,  each  circuit  is  to  be 
counted  as  positive  or  negative,  according  as  it  is  described 
in  the  direction  of  increasing  or  of  decreasing  angles. 

We  have  yet  to  call  attention  to  a  special  condition.  It  may 
happen  that  boundary-pieces,  which  belong  either  to  different 
boundary-lines  of  the  second  kind  or  to  one  such  line,  meet 
in  single  points  S.  In  such  cases  different  conceptions  are 
possible,  both  in  regard  to  how  a  boundary-line  shall  be  con- 
tinued beyond  a  point  S,  and  also  in  regard  to  the  connection 
of  the  surface-pieces  contiguous  to  S.  Now  formula  (8)  re- 
mains always  valid,  if  we  hold  a  conception  once  chosen.    Yet, 


240  THEORY  OF  FUNCTIONS. 

in  order  to  remove  all  difficulties  which  may  thereby  occur, 
and  in  order  to  have  something  definite,  we  will  assume  that, 
when  two  boundary-pieces  which  belong  to  boundary-lines  of 
the  second  kind  meet  in  a  point  S,  they  are  connected  by  an 
infinitely  small  cross-cut,  i.e.,  by  an  infinitely  small  boundaiy- 
line  of  the  first  kind.  The  advantage  is  thereby  secured,  that 
every  boundary-line  of  the  second  kind  without  exception,  if 
it  be  considered  by  itself,  that  is,  apart  from  boundary-lines 
of  the  first  kind  which  may  possibly  meet  it,  forms  a  simply 
closed  line. 

Formula  (6)  holds  quite  generally  for  surfaces  which  contain 
only  boundary-lines  of  the  first  kind.  Formula  (8)  on  the 
other  hand,  for  surfaces  which  possess  both  kinds  of  boundary- 
lines  or  only  those  of  the  second  kind,  holds  only  under  the 
condition  that  the  surface-pieces  which  are  removed  can  be 
extended  in  planes.  But  if  this  condition  be  satisfied,  then 
(8)  remains  equally  valid,  whether  or  not  T  itself  can  be  ex- 
tended in  a  plane.  We  will  emphasize  a  case  in  which  T  can 
be  so  extended,  and  in  Avhich  then  formula  (8)  can  be  again 
reduced  to  the  simple  relation  q  =  g  ~U+1.  If  we  assume 
that  the  complete  surface  W  is  closed  at  infinity,  and  if  the 
case  occur  in  which  all  the  n  points  at  infinity  have  been 
removed  from  T  by  means  of  boundary-lines  of  the  second 
kind,  which  together  make  n  circuits,  then  T  can  be  extended 
in  a  plane.  ^  If  this  case  occur,  the  outer  boundary-lines  make 
n  circuits,  and  the  other  V—  n  circuits  arise  from  the  inner 
boundary-lines.  The  latter  will,  according  to  the  hypothesis, 
be  so  described  that  the  pieces  B  which  are  removed  lie  on 
the  left,  and  T  therefore  on  the  right.  But  if  we  reverse  the 
direction  of  description,  in  order  to  establish  again  the  custom- 
ary hypothesis  that  T  lies  on  the  left,  each  circuit  at  the  same 
time  changes  its  sign,  and  consequently 

-(F-w)=w-F 

^  This  is  perhaps  the  only  case  in  which  T  itself  and  also  the  pieces 
which  were  removed  can  be  extended  in  planes ;  but  it  may  be  left  un- 
decided whether  this  cannot  occur  in  still  other  cases. 


SIMPLY  AND  MULTIPLY  CONNECTED   SURFACES.     241 

is  the  number  of  positive  circuits  for  these  boundary-lines. 
But  the  case  is  different  with  the  outer  circuits.  For  a  posi- 
tive circuit  (in  the  direction  of  increasing  angles),  in  a  piece 
which  is  removed  and  which  contains  a  point  at  infinity,  forms 
a  negative  circuit  in  T  when  that  surface  is  extended  in  a  plane. 
Therefore,  if  we  also  reverse  here  the  circuit-direction,  it  re- 
mains a  positive  circuit.  Thus  the  outer  boundary-lines  make 
n  positive  circuits,  the  inner  boundary-lines  n  —  V  such  cir- 
cuits, and  consequently  the  boundary-lines  of  the  second  kind 
contribute  2n-V  (9) 

to  U. 

This  value  is  increased  by  +1  by  every  cross-cut,  for  boun- 
dary-lines of  the  first  kind  (p.  229),  while  every  slit  leaves  it 
unchanged ;  for  a  change  of  direction  equal  to  -f  tt  occurs  at 
one  end  of  a  slit,  and  a  change  equal  to  —  tt  at  the  other. 
To  determine  the  number  of  cross-cuts,  which  are  contained 
only  in  boundary-lines  of  the  first  kind,  we  will  divide  these 
into  two  classes;  let  the  first  class  include  those  which  are 
connected  with  boundary-lines  of  the  second  kind,  the  second 
class  all  the  others.  The  values  of  e  and  K  which  refer  to 
these  two  classes  may  be  denoted  by  Ci  and  Ki,  and  eg  and  K^ 
respectively;  then 

e,  +  e2  =  e,  Ki  +  K,  =  K  (10) 

Let  us  keep  in  mind,  in  reference  to  the  first  class,  that 
when  the  sections  discussed  in  §  50  are  made  in  a  line-system, 
the  number  of  simple  line-segments  arising  is  always  the  same ; 
namely,  K«,  +  ^,), 

even  when  the  sections  are  so  effected  that  simply  closed  lines 
arise.  Hence  we  can  so  direct  the  sections  in  the  line-system 
under  discussion  that  all  the  boundary-lines  of  the  second  kind 
contained  in  it  become  simply  closed  lines.  Then  the  boundary- 
lines  of  the  first  kind  which  are  left  form  ^(ej  +  K^)  simple 
line-segments,  and  of  these,  since  all  the  e^  end-points  lie  in 

the  interior,  w     ,    r^  \  i  /  r^         \ 

'  i(ei  +  Ai)  -  ei  =  |(/ii  -  ei) 

are  non-dividing  cross-cuts. 


242  THEORY  OF  FUNCTIONS. 

The  second  class  of  boundary-lines  of  the  first  kind,  i.e.,  those 
which  are  not  connected  with  boundary -lines  of  the  second  kind, 
may  form  p  distinct  systems.  If  a  boundary-point  be  assumed 
on  each  of  the  latter,  they  form  by  (2) 

^(7^2  -  62)  +  p 

non-dividing  cross-cuts.  Since,  however,  every  boundary-point 
represents  a  negative  circuit,  they  furnish 

-  p  -f  [^(7^2  -  62)  +P^  =  i(/^2  -  62) 

positive  circuits.  Consequently  the  number  2n  —  V  found 
under  (9)  is  to  be  increased  by  ^{Ki  —  e^)  for  the  first  class 
of  boundary-lines  of  the  first  kind,  and  by  ^(TTg  —  e^  for  the 
second  class.    We  thus  obtain,  with  attention  to  (10),  the  value 

U=2n-V-\-\{K-e) 

for  the  number  U  of  positive  circuits  made  by  the  aggregate 
of  boundary-lines ;  by  means  of  this  relation  (8)  is  reduced  to 

From  the  results  of  this  paragraph  we  can  now  enunciate 
the  proposition : 

Every  Riemann  surface  which  possesses  only  a  finite  number 
of  sheets  and  brarich-points,  the  boundary-lines  of  which  form  a 
finite  line-system  (in  the  sense  of  §  50),  can  be  modified  into 
a  simply  connected  surface  by  means  of  a  finite  number  of 
cross-cuts. 

57.  We  will  now  conclude  these  investigations  by  making 
another  application,  namely,  to  the  determination  of  the  relation 
which  exists  between  the  mimber  of  corners,  edges,  and  faces  of 
an  arbitrary  body  bounded  by  plane  surfaces.^ 

If  we  denote  these  numbers  in  order  by  e,  Tc,  and  /,  then, 
according  to  a  proposition  by  Euler, 

e-k+f==2.  (1) 

1  F.  Lippich,  "Zur  Theorie  der  Polyeder,"  Sitz.-Ber.  d.  Wien.  Akad., 
Bd.  84,  Abth.  II.,  Juni-Heft,  1881. 


SIMPLY  AND  MULTIPLY  CONNECTED   SURFACES.     243 


Fig.  52. 


But  this  relation  does  not  hold  for  every  arbitrarily  formed 
body  with  plane  faces ;  on  the 
contrary,  a  number  which  de- 
pends upon  the  order  of  con- 
nection both  of  the  aggregate 
of  surfaces,  and  also  of  the 
individual  lateral  faces,  must 
in  general  be  added  to  the 
right  side.  For  instance,  the 
Eulerian  relation  does  not 
hold  for  the  body  represented 
in  Fig.  52,  in  which  a  smaller 
parallelopiped  so  rests  upon 
a  larger  that  the  face  of  the  smaller  covers  a  portion  of  the 
interior  of  a  face  of  the  larger.  We  can  at  once  convince  our 
selves  of  this  by  an  enumeration.  For  in  this  case  e  =  16, 
A;  =  24,  /=  11 ;  therefore 

e  —  k  +/=  3,  and  not  2, 

as  the  Eulerian  relation  requires.  In  like  manner  this  relation 
does  not  always  hold  if  there  be  a  cavity  in  the  body,  or  if  it 
be  closed  after  the  manner  of  a  ring. 

We  will  now  assume  that  the  aggregate  of  surfaces  of  the 
body  is  (q  +  l)-ply  connected ;  that  therefore  q  cross-cuts 
modify  it  into  a  simply  connected  surface.  Since  this  surface 
is  closed,  we  must,  by  §  46,  assume  a  boundary-point.  Let  this 
be  denoted  by  a,  and  be  situated  on  an  edge  (Fig.  52).  Now 
the  edges  form  a  line-system  in  the  surfaces  of  the  body.  This 
can  either  be  wholly  connected  or  consist  of  distinct  parts. 
Let  the  number  of  such  parts  be  n,  where  n  can  also  be  equal 
to  unity.  This  line-system  could,  by  §  51,  have  been  regarded 
as  a  system  of  cross-cuts,  if  it  had  satisfied  conditions  (1)  and 
(2),  given  in  that  paragraph.  Condition  (1)  is  indeed  satisfied, 
since  the  lines  possess  no  end-points  in  the  interior  of  the  sur- 
face ;  but  not  (2),  since,  in  case  n  be  not  equal  to  unity,  the 
lines  are  not  all  connected  with  the  boundary-point  a.  Never- 
theless we  can  cause  this  condition  to  be  satisfied,  by  also 


244  THEORY  OF  FUNCTIONS. 

assuming  one  boundary-point  on  an  edge  belonging  to  each  of 
the  other  n  —  1  parts  of  the  system  of  edges,  just  as  the 
boundary-point  a  was  assumed  in  one  of  those  parts.  Let 
these  points  be  denoted  by  a^,  a,,  ••-,  a„_i.  (In  Fig,  52  it  is 
necessary  to  assume  only  one  such  point,  aj.)  Then  the  sur- 
face possesses  n  boundary-points,  and  every  line  is  connected 
with  some  one  boimdary-point ;  consequently  condition  (2)  is 
satisfied.  Therefore,  by  §  53,  the  line-system  which  consists 
of  the  edges,  now  forms  a  system  of  cross-cuts,  quite  definite 
in  number ;  let  this  number  be  s. 

But  now,  after  n  —  1  new  boundary-points  are  taken  out  of 
the  surface,  its  order  of  connection  is  increased  by  n  —  1. 
Thus  q  -\-  n  —  1  cross-cuts  are  necessary  to  change  it  into  a 
simply  connected  surface.  If  we  imagine  the  surface  to  be 
cut  through  along  the  edges,  which  form  s  cross-cuts,  we 
resolve  it  into  distinct  pieces ;  namely,  into  the  individual 
bounding-faces  of  the  body,  the  number  of  which  was  /. 
These  are  not,  in  general,  all  simply  connected.  (In  Fig.  52 
one  was  not,  namely,  that  one  upon  which  the  smaller  body 
rests.)  If  we  denote  by .  p  the  total  number  of  cross-cuts 
which  are  necessary  to  make  all  the  bounding-faces  simply 
connected,  and  if  we  add  these  cross-cuts,  none  of  which 
divides  a  face,  we  again  obtain  /  distinct  pieces ;  these  pieces 
are  now,  however,  all  simply  connected.  Consequently  we 
have :  The  {q  +  w)-ply  connected  aggregate  of  surfaces  of 
the  body,  after  the  removal  of  the  n  —  1  boundary-points,  is 
resolved  hjs+p  cross-cuts  into  /  distinct  pieces,  each  of 
which  is  by  itself  simply  connected. 

But  now,  on  the  other  hand,  we  can  first  change  the  same 
surfaces  into  one  simply  connected  surface  by  means  of 
q  -\-n  —  1  cross-cuts,  and  then  resolve  this  surface  into  / 
distinct  pieces  by  means  of  /  —  1  additional  cross-cuts.  The 
former  surface  is  therefore  also  resolvable  into  /  distinct 
pieces,  each  by  itself  simply  connected,  by  means  of 

(q  +  n-l)  +  (f-l) 


SIMPLY  AND  MULTIPLY  CONNECTED   SURFACES.     245 

cross-cuts.  Consequently,  according  to  Riemann's  fundamental 
proposition, 

(^  +  n-l)  +  (/-l)  =  s+p, 

or  n-s+f=2+p-q.  (2) 

The  number  n  —  s  which  appears  in  this  formula  can  be 
expressed  in  terms  of  the  numbers  e  and  A:  of  corners  and 
edges.  For  at  every  corner  at  least  three  edges  meet,  and 
hence  each  corner  forms  a  nodal-point  of  the  line-system 
which  consists  of  these  edges  ;  and  if  we  denote  by  gg,  64, 
65,  •••  the  numbers  of  corners  in  which  3,  4,  5,  •••  edges  meet, 

we  have 

e  =  e3-f  e^-f  fisH . 

If,  further,  we  count  all  the  edges  which  meet  in  the  indi- 
vidual corners,  we  obtain  double  the  number  of  all  the  edges, 
since  every  edge  is  counted  twice.     Hence 

If  we  now  wish  to  resolve  the  system  of  edges  into  the 
s  cross-cuts  of  which  it  consists,  we  must  make  the  sections 
discussed  in  §  50 ;  then  the  s  cross-cuts  appear  as  s  simple 
line-segments,  the  number  of  which  is  half  as  great  as  the 
number  of  their  end-points.  Therein,  by  §  51,  each  of  the  n 
points  a,  ai,  Oo,  •••,  a„_i,  must  be  regarded  as  forming  two  end- 
points.  Since,  in  addition,  each  corner  in  which  h  edges  meet, 
as  an  ft-ple  nodal-point,  furnishes  h  —  2  end-points,  we  obtain 

2s  =  2n  +  es  +  2e,  +  3e,+  ": 

If  the  preceding  expression  for  2  A;  be  subtracted  from  this, 
we  get 

2s-2k  =  2n  —  2(63  -f  64  -f  65  H ) 

=  2n-2e; 

consequently  n  —  s  =  e  —  k, 

and  from  (2)  e-k+f=2+p-q.  (3) 


246  THEORY  OF  FUNCTIONS. 

This  is  the  desired  relation.  In  the  general  case,  therefore, 
the  number  p  —  q  is  added  to  the  number  2  on  the  right  side 
of  the  Eulerian  relation  (1) ;  in  this  q  is  the  number  of  cross- 
cuts necessary  to  modify  the  aggregate  of  surfaces  into  a  simply 
connected  surface,  and  p  the  number  necessary  to  that  end  in 
all  the  individual  boundary-faces. 

The  Eulerian  relation  therefore  holds  only  when  p  =  q.  In 
an  ordinary  polyedron,  everywhere  convex,  this  is  in  fact  the 
case,  because  then  p  =  g  =  0.  For  some  special  cases,  and  the 
way  in  which  the  numbers  e,  k,  f  must  be  counted  in  order 
that  equation  (3)  may  remain  valid,  we  refer  to  the  dissertation 
cited  above. 


SECTION  X. 

MODULI    OF    PEEIODICITY.^ 

58.  Let  f{z)  denote  an  arbitrary  algebraic  function.  Let  us 
conceive  as  the  region  of  the  variable  z  a  surface  consisting 
of  as  many  sheets  and  containing  such  branch-points  as  the 
nature  of  this  function  f{z)  requires.  We  will  surround  with 
small  closed  lines  the  points  of  discontinuity  of  this  function 
and  thus  exclude  them.  We  will  assume  provisionally  that 
all  the  points  of  discontinuity  are  enclosed  in  this  way,  but  we 
shall  very  soon  see  that  certain  kinds  of  points  of  discontinuity 
need  not  be  excluded.  We  will  call  the  surface  so  formed  T. 
This  now  possesses  a  finite  order  of  connection,  and  can  there- 
fore, if  it  be  multiply  connected,  be  modified  into  a  simply 
connected  surface  by  means  of  a  finite  number  of  cross-cuts. 
For,  since  the  function  in  question  is  an  algebraic  one,  this  is, 
by  §  54,  at  all  events  the  case  before  the  exclusion  of  the 
points  of  discontinuity.  But  since  an  algebraic  function  pos- 
sesses only  a  finite  number  of  points  of  discontinuity  (§  38), 

^  The  special  investigation  of  the  logarithmic  and  exponential  functions 
given  in  §  22  and  §  23  may  serve  as  illustrations  of  the  general  considera- 
tions contained  in  this  section.    Other  examples  will  be  found  in  §  61. 


MODULI  OF  PERIODICITY.  247 

therefore  by  the  exclusion  of  these  points  only  a  finite  number 
of  new  boundary-lines  are  added;  accordingly  by  §  56  the 
order  of  connection  also  remains  finite  after  the  exclusion  of 
the  points  of  discontinuity.  Consequently,  if  the  surface  T 
be  multiply  connected,  we  will  modify  it  into  a  simply  con- 
nected surface  by  means  of  cross-cuts,  and  designate  the  new 
surface  by  T'.  Then  every  closed  line  in  T'  forms  the  com- 
plete boundary  of  a  portion  of  the  surface,  in  which  /(z)  is 
finite  and  continuous.  Hence,  if  the  function  defined  by  the 
integral 


be  formed  by  integrating  from  an  arbitrary  fixed  initial  point 
Zq  to  a  point  z,  along  an  arbitrary  path  which  lies  wholly 
within  T\  then  any  two  such  paths  together  make  a  closed 
line,  and  this  line  bounds  completely  a  portion  of  the  surface 
in  which  f{z)  is  everywhere  continuous ;  therefore  w  acquires 
at  z,  along  all  such  paths,  one  and  the  same  value  (§  18). 
Consequently  w  is  a  function  of  the  upper  limit  z,  and  remains 
uniform  everywhere  within  T'} 

1  The  case  in  which  two  paths  taken  together  form  a  closed  line  which 
intersects  itself  is  no  exception  to  the  above.  For  we  can  always  resolve 
such  a  line  into  several  simply  closed  lines.  (Cf.  Fig.  53.)  The  resolu- 
tion is  effected  in  the  following  way : 
Whenever,  in  tracing  the  line  from 
an  arbitrary  point  z^,  we  have  returned 
to  a  point  already  once  passed  {e.g., 
a),  and  thus  have  traced  a  simply 
closed  line  {e.g.,  abcda),  we  separate 
this  and  regard  the  part  which  follows 
(e.g.,  ae)  as  the  continuation  of  the 
part  {Zfja)  which  preceded  the  part 
separated.  If  this  mode  of  procedure 
be  repeated  as  often  as  the  same  con- 
dition arises,  there  is  finally  left  a 
line  likewise  simply  closed ;    and  in  pjg  53 

this  way  the  given  line  is  resolved  into 

several  simply  closed  lines.  (In  the  figure  the  lines  which  are  separated 
are  ahcda  and  efghe,  and  that  which  is  left  is  ZQaeifhbdzQ.)     The  above 


248 


TUEOBY  OF  FUNCTIONS. 


But  the  case  is  different  when  we  consiaer  the  function  lo 
in  the  surface  T,  and  when  therefore  we  let  the  path  of  inte- 
gration cross  the  cross-cuts.  In  order  to  examine  this,  we 
will  first  direct  our  attention  to  the  case  in  which  no  cross-cut 

is  divided  into  segments  by  a  sub- 
sequent one  which  starts  from  it. 
Now  both  edges  of  each  cross-cut 
belong  to  the  boundary  of  T',  so 
that  these  are  connected,  and  we 
can  draw  a  closed  line  h,  running 
entirely  in  the  interior  of  T',  which 
leads  from  one  edge  of  the  cross- 
cut to  the  other  edge  of  the  same. 
Let  Zi  and  z^  (fig.  54)  be  two 
points  lying  infinitely  near  each 
other  on  opposite  sides  of  the 
cross-cut.     We  will  now  inquire  whether 

when  the  paths  of  integration  still  run  entirely  in  T',  acquires 
at  Zi  and  z^  values  that  are  equal  (accurately  speaking,  differ- 
ent by  an  infinitesimal  quantity)  or  different.  But  if  we 
denote  the  values  of  w  at  z^  and  z^  by  w^  and  w^  respectively, 
we  have 


=  Cf{z)dz=  r\f(z)dz+  rf{z)dz, 

Jz^  Jz^  Jz^ 


the  first  integral  to  be  taken  along  an  arbitrary  path  running 
in  T\  the  second  along  a  closed  line  h  leading  from  z^  to  z^ 
within  T.     Thus 


Wg  —  Wi  =  I   'f{z)dz 


integral,  extended  along  the  simply  closed  lines,  is  now  equal  to  zero,  and 
therefore  it  is  also  zero  taken  along  the  given  line,  since  this  integral 
is  equal  to  the  sum  of  the  preceding.  Then,  if  the  given  path  be  formed 
by  two  paths  leading  from  z^  to  z,  the  integral  has  the  same  value  along 
both  paths  (§  18). 


MODULI  OF  PERIODICITY.  249 

Hence  ici  and  w^  have  the  same  or  different  values  according 
as  the  integral 


ff(z)dz, 


extended  along  the  closed  line  b,  is  zero,  or  has  a  value  A  dif- 
ferent from  zero.  In  the  first  case  w  remains  continuous  on 
crossing  the  cross-cut ;  in  the  latter  case  w  springs  abruptly  from 
Wi  to  Wo  =  Wi-\-A,  and  is  therefore  discontinuous.  But  this 
abrupt  change  is  the  same  at  all  places  of  the  same  cross-cut, 
because  the  value  of  the  integral  does  not  change,  if  we  enlarge 
or  contract  the  closed  line  b  in  such  a  way  that  it  begins  and 
ends  at  two  other  infinitely  near  points  on  opposite  sides  of 
the  same  cross-cut  (§  19).  This  quantity  A,  which  is  thus 
constant  along  the  entire  cross-cut,  and  by  which  the  function- 
values  on  one  side  of  the  cross-cut  exceed  those  on  the  other, 
is  called  the  modulus  of  periodicity  corresponding  to  this  cross- 
cut. The  case  is  exactly  similar  for  every  cross-cut,  because 
the  two  edges  of  each  one  are  connected,  and  therefore  a  closed 
line  can  be  drawn  from  a  point  on  one  side  to  an  infinitely 
near  point  on  the  other  side  through  the  interior  of  T'.  Thus 
to  every  cross-cut  corresponds  a  modulus  of  periodicity,  which 
remains  constant  for  one  and  the  same  cross-cut  (yet  always 
under  the  hypothesis  that  no  cross-cut  is  divided  into  segments 
by  a  subsequent  one).  But  if  we  now  assume  that  the  function 
tv  proceeds  continuously  in  T  also,  and  hence  also  over  the 
cross-ciit,  it  acquires  at  Zj,  on  the  path  z^^z.^i,  which  crosses 
the  cross-cut,  a  value  greater  by  the  modulus  of  periodicity 
than  the  value  acquired  on  the  path  ZqZi,  which  does  not  cross 
the  cross-cut.  For  in  the  former  case  the  value  of  xc  at  z^  is 
regarded  as  the  uninterrupted  continuation  of  Wg,  while  on  the 
second  path  w  acquires  the  value  Wi,  and 

Wg  =  Wi  4-  A. 

There  occurs  here  a  condition  similar  to  that  which  we  found 
to  exist  in  the  case  of  branch-cuts  (cf.  §  13),  and  as  long  as  the 
surface  T  consists  of  only  a  single  sheet,  we  can  also  regard 
every  cross-cut  as  actually  a  branch-cut,  over  which  the  surface 


260 


THEORY  OF  FUNCTIONS. 


continues  into  another  sheet.  But  we  must  then  suppose  that 
infinitely  many  sheets  lie  one  below  another,  since,  for  every 
new  passage  of  the  cross-cut,  the  value  of  the  function  iv  is 
increased  by  A,  and  the  original  value  never  occurs  again. 
If  the  surface  T  itself  already  consist  of  several  sheets,  that 
mode  of  representation  would  indeed  be  possible,  but  yet  it 
would  be  too  complicated,  and  hence  would  offer  no  real 
advantage. 

The  sign  of  A  changes  if  the  closed  line  b  be  described  in 
the  opposite  direction;  but  we  will  always  so  assume  the 
modulus  of  periodicity  that  it  is  equal  to  the  integral  taken 
along  the  closed  line  &  in  the  direction  of  increasing  angles. 

If  we  now  conceive  all  possible  paths  which  lead  from  an 
initial  point  Zq  to  an  arbitrary  point  z  through  the  interior  of 
T,  then  these  paths  can  either  cross  none  of  the  cross-cuts  or 
intersect  one  or  more  cross-cuts  one  or  more  times.  Hence  w 
can  acquire  at  one  and  the  same  point  z  very  different  values, 
according  to  the  nature  of  this  path,  and  it  is  therefore  a 


Fio.  55. 


multiform  function  of  the  upper  limit  of  the  integral.  But 
since  this  diversity  of  values  of  iv  at  the  point  z  is  due  solely 
to  the  passages  over  the  cross-cuts,  these  different  values  can 


MODULI  OF  PERIODICITY.  251 

differ  from  one  another  only  by  multiples  of  the  moduli  of 
periodicity.  Hence,  if  Ai,  A.^,  A^,  •••  denote  the  moduli  of 
periodicity  for  the  single  cross-cuts,  nj,  n2,  n^,  •  •  •  positive  or  nega- 
tive integers,  and  lo  and  w'  two  different  values  of  w  at  the 
point  z,  then 

w'  =  10  +  riiAi  +  «2^2  +  risAs  H . 

An  example  may  make  this  clear.  Fig.  55  represents  a 
triply  connected  surface ;  let  the  cross-cuts  be  ab  and  cd,  and 
let  the  moduli  of  periodicity  for  the  same  be  Ai  and  Az,  re- 
spectively, so  taken  that  the  passage  from  one  side  of  the 
cross-cut  to  the  other  side  along  a  closed  line  is  made  in  the 
direction  of  increasing  angles.  If  we  designate  the  value 
acquired  by  the  function  w  on  a  path  by  adding  the  path  in 
brackets  to  the  letter  w,  we  have 

w(zoez)  =  w(ZftZ)  +  As, 
^(zofgz)  =  H^(^)  —A1  +  A2, 
w(zjiz)  =  w(Z(;z)  +  Ai. 

Erom  this  it  is  evident  that  the  function  defined  by  the 
integral 

possesses  a  multiformity  of  a  quite  peculiar  kind ;  namely,  that 
the  different  values  which  it  can  acquire  for  the  same  value  of 
z  differ  from  one  another  only  by  multiples  of  constant  quanti- 
ties. If  we  now  take  the  inverse  function,  i.e.,  if  we  regard  z 
as  a  function  of  w,  then  this  is  a  periodic  function,  since  it 
remains  unchanged  when  we  increase  or  diminish  the  argument 
w  by  arbitrary  multiples  of  the  moduli  of  periodicity.  By  this 
also  the  name  modulus  of  periodicity  is  justified,  since  we  can 
say,  analogously  to  the  language  of  the  theory  of  numbers,  that 
z  acquires  equal  values  for  such  values  of  w  as  are  congruent 
with  one  another  to  a  modulus  of  periodicity,  i.e.,  as  have  a 
difference  equal  to  a  multiple  of  the  modulus  of  periodicity. 

59.  We  have  hitherto  assumed  that  the  cross-cuts  are  so 
drawn  that  no  one  of  them  is  divided  into  segments  by  a  subse- 


252 


THEORY  OF  FUNCTIONS. 


quent  cross-cut  which  starts  from  it.  But  if  one  be  so  divided, 
as  for  instance  in  Fig.  56,  where  the  one  cross-cut  ad  is  divided 
by  the  second  ce  into  the  two  segments  ac  and  cd,  the  modulus 
of  periodicity  Bi  of  the  one  segment  ac  may  possibly  diiler 
from  that  B2  of  the  other  segment  cd.     For  Bi  is  equal  to  the 

integral    j  f{z)dz  taken  along  the  line  61,  B^  is  equal  to  the  same 

integral  taken  along  62-  If  these  integrals  have  different  values, 
then  the  moduli  of  periodicity  B^  and  B^  are  different.     Thus 


Fig.  .')(•.. 

the  modulus  of  periodicity  does  not  now  remain  constant  along 
an  entire  cross-cut,  but  only  from  one  node  of  the  net  of  cuts 
to  the  next.  But  now  a  modulus  of  periodicity  ^3  corresponds 
to  the  cross-cut  ce,  and  hence  there  are  three  moduli  of  perio- 
dicity, notwithstanding  that  only  two  cross-cuts  are  necessary 
to  modify  our  surface  into  a  simply  connected  surface.  But  in 
such  a  case  there  always  exist  relations  between  the  single 
moduli  of  periodicity.  In  our  example  the  integral  taken  along 
63  is  equal  to  the  sum  of  the  integrals  taken  along  hi  and  62 
(§  19),  and  hence 

B,=  B,  +  B,; 

thus  we  have  in  fact  only  two  moduli  of  periodicity  which  are 


MODULI  OF  PERIODICITY.  253 

independent  of  each  other,  i.e.,  just  as  many  as  there  are  cross- 
cuts. 

To  prove  now  in  general  that  there  are  always  only  as  many 
moduli  of  periodicity  independent  of  one  another  as  there  are 
cross-cuts,  we  observe  that  the  cross-cuts  in  most  cases  can  be 
drawn  in  various  ways.  But  there  is  always  o^f^  TTIP^^  ^^ 
resolution  in  which  no  cross-cut  is  divided  into  segments  by  a 

subsequent  cross-cut. This  is  always  effected  by  beginning 

every  cross-cut  at  a  j^pint  of  the  original  boundary  and  also 
ending  it  at  such  a  point.  If  the  surface  be  closed  and  hence 
possess  only  a  single  boundary-point  (§  46),  we  have  only  to 
begin  and  end  each  cross-cut  at  this  point. 

Now  let  an  (?i-f-l)-ply  connected  surface  first  be  so  resolved 
into  a  simply  connected  surface  by  means  of  n  cross-cuts  that 
thereby  no  cross-cut  is  divided  into  segments  by  another ;  we 
then  have,  for  this  mode  of  resolution,  exactly  as  many  moduli 
of  periodicity  as  cross-cuts.     Let  these  be 

-4l,  Ao,   •••,   An. 

Next  let  the  same  surface  be  resolved  in  another  arbitrary  way. 
Thereby  the  single  cross-cuts  are  divided  into  segments  with 
different  moduli  of  periodicity,  and  the  number  of  the  latter  is 
greater  than  n ;  let  these  be 

Now  let  the  variable  z  describe  from  any  arbitrary  point  Zq  a 
closed  line  which  crosses  only  one  cross-cut  of  the  first  system, 
and  let  the  modulus  of  periodicity  for  this  cross-cut  be  A^; 
then,  if  Wq  and  w  denote  the  values  of  the  function  at  the 
beginning  and  after  the  completion  of  the  closed  line,  we  have 

w  =  zvq  +  A^. 

But  if  we  now  suppose  the  surface  to  be  resolved  in  the  second 
way,  the  same  closed  line  may  cross  several  cross-cuts  of  the 
second  system  ;  hence  by  §  58  the  value  of  w  must  be  obtained 
also  in  the  form 


254  THEORY  OF  FUNCTIONS. 

wherein  h  denotes  a  positive  or  negative  integer  (zero  included). 
Consequently 

A,  =  hiBi  +  h,B,  +  •  • .  +  h„,B^. 

Now,  conversely,  let  the  variable  z  describe  from  z^  a  closed 
line  which  crosses  only  one  cross-cut  of  the  second  system, 
and  let  the  modulus  of  periodicity  of  this  cross-cut  be  B^^^ ;  then 
the  final  value  of  the  function  is  first 

iVo-\-B^; 

but,  if  the  crossings  of  the  cross-cuts  of  the  first  system  be 
considered,  that  value  is  also  obtained  in  the  form 

1^0  +  9iA  +  QiA  -\ 1-  gnA, 

wherein  g  likewise  denotes  a  positive  or  negative  integer  (zero 
included).     From  this  follows 

Consequently  we  obtain  between  the  two  systems  of  the 
moduli  of  periodicity  A  and  B  the  following  two  sets  of 
equations : 

A,  =  h,"B,    +  Ju"B,  +•••+  JiJ'B^ 


(1) 
and 

(2) 


Bi  =  gMi    +  gM^   -\ f-  gJA 

B^  =  gi'A,   +  g,'A^  4-  -  4-  gJ'A 

B^  =  g,^-KA,  +  g.^^KA,  +  -  -f  g^^^^A^. 


Since  now  according  to  the  assumption  m  >  n,  we  can 
eliminate  the  n  quantities  A  from  equations  (2)  and  thereby 
obtain  m  —  n  relations  between  the  quantities  B.  But  since 
we  can  also  obtain  these  relations  by  substituting  in  (2)  the 
values  of  A  from  (1),  they  must  be  homogeneous  linear  equa- 
tions with  integral  coefficients.  Therefore  we  conclude:  If 
previous  cross-cuts  be  divided  by  subsequent  cross-cuts  into 
segments  which  have  different  moduli  of  periodicity,  so  that 


MODULI  OF  PERIODICITY. 


255 


in  all  m  moduli  of  periodicity  exist,  while  only  n  cross-cuts 
occur,  then  there  are  m  —  n  linear  homogeneous  equations  of 
condition  with  integral  coefficients  between  these  m  moduli 
of  periodicity,  and  of  these  moduli  only  n,  i.e.,  just  as  many 
as  there  are  cross-cuts,  are  independent  of  one  another. 

We  can  also,  without  any  calculation,  reach  the  same  con- 
clusion by  a  simple  consideration.  For,  after  the  surface  has 
been  made  simply  connected  by  means  of  cross-cuts,  its  boun- 
dary can  be  traced  in  a  continuous  description  (§  53,  IX.).  The 
cross-cuts  and  their  segments  enter  in  this  description  in  a 
definite  succession.  If,  for  each  cross-cut,  the  modulus  of 
periodicity  be  known  for  that  segment  at  which  we  arrive 
first  in  the  description,  then  the  moduli  of  periodicity  for  the 
other  segments  are  given  by  linear  relations.  We  will  show 
this  only  in  an  example. 

In  the  quadruply  connected  surface  represented  by  Fig.  57, 
let  ab,  cd,  ef  be  the  three  cross-cuts  which  modify  the  surface 
into  a  simply  connected  sur- 
face. Let  the  letters  p,  q,  r, 
s,  t,  u,  V,  X,  y,  z  denote  the 
values  acquired  by  the  func- 
tion w  at  the  corresponding 
points  which  are  situated  in- 
finitely near  the  cross-cuts.  If 
we  now  describe  the  cross-cuts, 
together  with  the  original 
boundary,  in  the  direction 
aefc  •■',  let  the  moduli  of 
periodicity  be  known  for  the 
three  segments  ae,  ef,  fc,  and 
be  denoted  by 


Fig.  57. 


q  —p  =  s  —  r  =  Ai,  s  —  u  =  x  —  v  =  Ao,  x  —  y  =  A3', 

we   then   wish  to   obtain   the   moduli  of  periodicity  for  the 
segments  eb  and  fd,  and  will  denote  these  by 


u  —  t  =  Xi,  z  —v=  X2. 


266  THEORY  OF  FUNCTIONS. 

To  find  these,  we  remark  that  continuity  exists  between  the 
function-values  at  any  two  consecutive  points  which  are  not 
separated  by  a  cross-cut ;  that  their  difference  is  therefore  in- 
finitesimal.    Consequently  we  can  let 

Thus  we  obtain  '         ^ 

Xi  =  u  —  t  =  u  —  r  =  (s  —  r)  —  {s  —  u)=  Ai  —  A^, 
Xz  =  z  -  V  =  y  —  V  =  {x  —  v)-{x  —  y)  =  A^  —  A^, 
by  which  Xj  and  Xg  are  expressed  in  terms  of  A-^,  A2,  Ag. 

60.  We  have  hitherto  assumed  that  all  the  points  of  discon- 
tinuity are  removed  from  the  2!-surface  by  means  of  small 
enclosures,  so  that  the  function  f(z)  remains  finite  in  the  sur- 
face T  so  formed.  But  we  will  now  show  that  it  is  in  fact  not 
necessary  to  exclude  all  the  points  of  discontinuity,  and  will 
inquire  for  what  points  the  enclosures  need  not  be  drawn. 

The  modulus  of  periodicity  A  for  a  particular  cross-cut  is,  as 

was  shown  in  §  58,  the  value  of  the  integral    |  f(z)dz,  extended 

over  a  closed  line  b  which  leads  from  one  side  of  the  cross-cut 
through  the  interior  of  the  simply  connected  surface  T'  to  the 
other  side  of  the  same  cross-cut.  But  this  integral  in  many 
cases  may  have  the  value  zero.  Let  us  assume  that  the  closed 
line  h  encloses  a  place  removed  from  the  z-surface  which  con- 
tains a  point  of  discontinuity  a  (which  is  not  at  the  same  time 
a  branch-point)  of  the  function  f{z).     Then  by  §  42  the  integral 

I  f(z)dz  has  a  value  different  from  zero  only  when  the  term 


z  —  a 
is  present  in  the  expression  which  indicates  how  f(z)  becomes 
infinite ;  in  all  other  cases  the  integral  has  the  value  zero.     For 
instance,  the  integral  equals  zero  when  f(z)  is  infinite  at  a  as 


{z  —  a)'-' 
or  as 

An)  f,(n+l) 

— 1 h 

(z  -  a)"      {z-  a)"+^ 


MODULI  OF  PERIODICITY.  257 

is  infinite,  wherein  71  denotes  a  positive  integer  different  from 
unity.  In  such  a  case  the  function  w  remains  continuous  on 
crossing  a  cross-cut ;  hence  it  is  not  necessary  to  exclude  the 
point  of  discontinuity,  and  the  cross-cut  need  not  be  considered. 
If  we  assume,  for  instance,  a  simply  connected  piece  of  the 
2-surface,  in  which  are  contained  only  points  of  discontinuity 
of  the  kind  in  question,  then  the  integral  |  f(z)dz  acquires  the 
same  value  along  two  paths  which  enclose  such  a  point  of  dis- 
continuity, because  this  integral,  taken  round  the  point  of 
discontinuity,  has  the  value  zero  (§  18).  Hence,  in  such  a 
piece  of  the  surface,  the  function 


w 


=  Cf{z)dz 


is  likewise  a  uniform  function  of  the  upper  limit,  just  as 
if  the  piece  of  the  surface  contained  no  point  of  disconti- 
nuity at  all. 

This  is  one  kind  of  point  of  discontinuity  which  need  not 

be  excluded.     Let  us  now  turn  to  branch-points.     The  integral 

\jXz)dz,  taken  along  the  closed  line  6,  has   the  value  zero 

when  this   line    encloses   a  winding-point  of  the   (m  —  l)th 

order  at  which  f{z)  becomes  infinite  of  an  order  not  higher 

^■^1 ]|_ 

than  (§  21)  ;  and,  in  general,  when  the  term  which  is 

infinite  of  the  first  order  is  Avanting  in  the  expression  which 
indicates  how  f{z)  becomes  infinite  at  the  branch-point  (§  42). 
In  this  case,  therefore,  the  discontinuity-  and  branch-point 
need  not  be  exchided,  and  thus  it  is  likewise  unnecessary 
to  consider  the  cross-cut.  But  we  remark  that,  since  the 
z-surface  now  consists  of  several  sheets,  it  may  be  multiply 
connected  without  the  exclusion  of  points  of  discontinuity. 
Thus  cross-cuts  will  always  in  such  cases  be  required  in  order 
to  modify  the  surface  into  a  simply  connected  surface,  and  to 
these  will  correspond  moduli  of  periodicity. 

Finally,  we  can  also  determine  in  what  case  the  point  at 
infinity  must  be  excluded.     The  value  of  the  integral,  for  a 


258  THEORY  OF  FUNCTIONS. 

line  enclosing  the  point  2  =  go,  depends  upon  the  nature  of  the 
function 

for  2  =  00  (§  43).     Thus  this  point  must  be  excluded  when 

lim  [2/(2;)]^^  is  finite,  and  not  zero; 

and  in  general  when,  and  only  when,  in  the  development  of 
f(z)  in  ascending  and  descending  powers  of  z,  a  term  of  the 
form 

9 

z 
is  present. 

If  now,  for  a  given  function  f(z),  all  those  points  have  been 
excluded  from  the  ^-surface  which  must  necessarily  be  excluded, 
and  only  these,  then,  ivithin  the  surface  T  so  formed,  the  integral 

I  f{z)dz,  taken  along  a  dosed  line  which  forms  by  itself  alone  the 

complete  hoxmdary  of  a  portion  of  the  surface,  is  always  equal 
to  zero. 

For  the  portion  of  the  surface  so  bounded  contains  then 
either  no  points  of  discontinuity  at  all,  or  only  such  as  lead 
to  the  value  zero  for  the  integral  taken  along  the  boundary. 
In  this  it  is,  of  course,  assumed  that  the  closed  line  does  not 
pass  through  a  jsoint  of  discontinuity  or  a  branch-point. 

61.  We  will  now  apply  the  preceding  considerations  to  some 
examples. 

1.    Tlie  Logarithm. 

We  will  recall  first  the  function  log  z,  or  the  function  defined 
by  the  integral 

J'^'dz 
.    7' 

already  discussed  in  §  22  and  §  23.    In  this  ^2;)  =  -  is  uniform, 

and  hence  the  ^-surface  consists  of  one  sheet.  Further,  «  =  0 
is  a  point  of  discontinuity,  and 

lim izf{z)']^  =  lim [z  . -\     =  1. 


MODULI  OF  PERIODICITY. 


259 


Hence  this  point  must  be  excluded.     If  we  now  assume  that 

the  2;-surface  is  closed  at  infinity,  the  point  2  =  co  must  also  be 
excluded,  because 

lim[2/(z)],^=l. 

By  the  exclusion  of  these  two  points,  the  surface  T  is  made 
doubly  connected,  and  a  cross-cut  which  connects  the  circles 
enclosing  the  two  points  0  and  00  modifies  it  into  a  simply  con- 
nected surface  (Fig.  58). 
The  modulus  of  perio- 
dicity A  is  equal  to  the 
value  of  the  integral 

-0 


rdz 


00 


Fig.  58. 


taken  along  a  closed  line, 
which  makes  a  circuit 
round  the  origin  in  the 
direction  of  increasing  angles,  and  hence 

^  =  2  Trt. 


Such  a  line  also  encloses  the  point  00  at  the  same  time,  and 
for  this  we  obtain  (§  43) 


/? 


=  —  2  TT?'  lim 


z^  ~ 
z 

~z~ ) 


—   '1  TTl, 


if  the  integration  be  extended  in  the  positive  boundary  direc- 
tion, and  if,  therefore,  the  cross-cut  be  crossed  in  a  direction 
opposite  to  the  former. 


2.    The  Inverse  Tangent. 

dz 


Here 


Jo  1 


+  z^ 


1+z^ 


260  THEORY  OF  FUNCTIONS. 

is  likewise  uniform  and  becomes  infinite  of  the  first  order,  for 
z  ■-=  i  and  z  =  —  i ;  on  the  other  hand, 


lim  [2/(2;)  ]^^«,  =  liii^ 


1  +  z' 


=  0. 


Hence  we  need  exclude  only  the  points  z  —  i  and  z  =—  i  by 
means  of  small  circles  (Fig.  59),  and  we  then  obtain,  assuming 

the  2!-surface  to,  be  closed  at  infinity,  a  doubly  con- 
f-u    nected  surface;  this  is  changed  into  a  simply  connected 

surface  by  a  cross-cut  Avhich   joins   the   small   circles 

round  +  i  and  —  i.     The  modulus  of  periodicity  A  is 

the  value  of  the  integral 


/' 


dw, 


0 

Fig.  59. 


taken  along  a  closed  line  which  makes  a  circuit  round 
the  point  +  i  in  the  direction  of  increasing  angles,  and 
hence,  as  we  have  already  found  in  §  20, 

A  =  7r. 


The  same  line  can  be  regarded  as  one  which  makes  a  circuit 
round  the  point  —  i  in  the  direction  of  the  decreasing  angles, 
and  it  then  furnishes  the  same  modulus  of  periodicity. 

If  we  now  assume  that  the  2!-surface  is  not  closed  at 
infinity,  but  is  bounded  by  a  closed  line  which  we  then  enlarge 
indefinitely,  the  surface  T  becomes  triply  connected  when  the 
two  points  +  *  and  —  i  are  excluded.  Therefore,  two  cross- 
cuts are  in  this  case  necessary  to  change  the  surface  into  one 
simply  connected.     But  now,  since  the  integral 


A 


dz 


+  z^ 


taken  along  a  closed  line,  has  the  value  +  tt  or  —  tt  or  0, 
according  as  the  line  makes  a  circuit  round  +  i  or  —  i  or  both, 
in  the  direction  of  increasing  angles  (§  20),  the  moduli  of 
periodicity  in  reference  to  the  two  cross-cuts  have  the  values 
-f  TT  and  —  TT,  or  the  one  has  the  value  ±  ir  and  the  other  the 
value  zero,  according  to  the  mode  of  drawing  the  cross-cuts. 


MODULI  OF  PERIODICITY. 


261 


Hence  the  function  iv  =  arc-tan  z  also  changes  here  by  mul- 
tiples   of  IT. 

The  inverse  function  z  =  tan  w  is  now  periodic  with  the 
period  tt.  The  representation  of  the  ^-surface,  assumed  to  be 
closed  at  infinity,  on  the  ic-surface,  is  here  made  in  a  way 
exactly  similar  to  that  shown  in  §  23  for  the  exponential  func- 
tion ;  in  place  of  the  circles  enclosing  the  points  0  and  oo  there 
enter  here  only  those  which  enclose  the  points  +  i  and  —  *■. 
If  we  assume  that  the  cross-cut  wliich  joins  these  circles  runs 
along  the  ordinate  axis,  the  w-surface  is  divided  into  strips 
bounded  by  straight  lines  which  rvm  parallel  to  the  ordinate 
axis,  and  which  pass  through 
the  points  0,  ±  ir,  ±  2  tt, 
±377,  ...  (Fig.  60).  In  each 
of  these  strips  the  function 
z  =  tan  w  acquires  all  its 
values,  and,  indeed,  each  but . 
once,  because,  except  as  to 
multiples  of -the  modulus  of 
periodicity,  only  one  value  of 
IV  corresponds  to  each  value 
of  z,  the  z-surface  consisting 
of  only  one  sheet. 

We  will  now  examine  this  function  in  the  inverse  manner, 
by  commencing  with  the  periodic  function.  If  2;  =  <l>(w) 
denote  a  uniform  simply  periodic  function  with  the  modulus 
of  periodicity  A,  that  is,  a  uniform  function  which  possesses 
the  property  that 

<f>(w  +  A)  =  cl>(iv), 

then  the  tc-surface  can  be  so  divided  into  strips  that  the  func- 
tion acquires  all  its  values  in  each  strip,  and  has  the  same  value 
at  every  two  points  situated  in  different  strips  which  differ  by 
A  or  a  multiple  of  A  (Fig.  61).  For,  if  we  draw  any  line 
BC  which  does  not  intersect  itself,  the  points  w  +  A,  which 
are  obtained  from  the  points  of  the  line  BC  by  adding  A, 
form  a  line  DE  parallel  to  the  line  BC.     Thus  the  function  <f> 


Fig.  60. 


262 


THEOBT  OF  FUNCTIONS. 


F- 


Pig.  61. 


has  the  same  values  along  DE  as  along  BC.  The  same  is 
true  of  all  lines  which  run  parallel  to  these  at  equal  dis- 
tances. Moreover,  if  w 
be  a  point  in  the  inte- 
rior of  the  strip  BCDE, 
then  w  -\-  A  lies  in  the 
interior  of  the  adjacent 
strip  DEFG,  w  +  2A 
in  the  interior  of  the 
next  following  strip, 
etc.  Hence  at  these 
points  the  function 
again  has  the  same 
value.  Now,  since 
every  two  points,  w 
and  w  -f  nA,  at  which  the  function  has  the  same  value, 
lie  in  different  strips,  it  must  acquire  all  its  values  in  each 
strip. 

We  will  now  assume  further  that  the  function  z  =  <\>{w) 
becomes  infinite  of  the  first  order  at  only  one  finite  point  w  =  r 
in  one  and  the  same  strip;  we  can  then  show  that  it  also 
becomes  zero  only  once  in  every  strip  and  hence  acquires  each 
value  only  once.  To  this  end  let  the  points  at  which  <f)(w) 
becomes  zero  within  the  strip  considered  be  denoted  by 
s,  s',  s",  .  .  .,  and  let  the  number  of  these  points  be  n  and  assume 
that  none  of  them  lies  at  infinity.  If  we  now  draw,  from  two 
points  w  and  lo  +  c  situated  on  one  of  the  two  lines  which  bound 
the  strip,  straight  lines  to  the  points  w  -\-  A  and  iv  -\-  c  +  A, 

situated  on  the  other  bounding- 
line  (Fig.  62),  we  obtain  a  par- 
allelogram with  vertices  w,w-\-c, 
w -\- c -\- A,  w  +  A;  and  if,  as 
was  assumed,  the  points  r,  s, 
s',  s", ...  all  lie  in  the  finite  part 
of  the  surface,  we  can  always 
so  choose  the  points  w  and  w-\-c 
lie  within  the  parallelogram.     If  we  now 


_w-{-A 


w+c+A 


w  +  c 


Fig.  62. 


that 


MODULI  OF  PERIODICITY.  263 

take  the  integral  I  d  log  4>(^w)  along  the  boundary  of  this  par- 
allelogram, we  obtain  by  §  35,  (1), 

I  d  log  4>(w)  =  2  Tri(n  —  1), 

since  4'{io)  becomes  n  times  zero  and  once  infinite  within  the 
parallelogram.  This  integral  may  be  divided  into  four  parts, 
taken  along  the  four  sides  of  the  parallelogram.  But  we 
remark  that   |  d  log  <l>(w)  is  independent  of  the  path  of  integrar 

tion  as  long  as  this  does  not  cross  one  of  the  lines  rs,  rs',  etc., 
each  of  which  connects  points  at  which  <f){w)  becomes  infinite 
or  zero  (§  22).  If  we  take  it  along  the  straight  line  which 
leads  from  to  -\-  A  to  w,  it  acquires  the  value  zero ;  for  in  the 
first  place  it  is  equal  to  log^(^(?)  —  log  <^(w -|- ^),  and  since 
none  of  the  lines  rs  is  crossed,  not  only  is  ^(w  -{-  A)  =  <l>{w), 
but  also  log  ^(lu  +  A)=  log  (f)(w).  [If  one  line  rs  were  crossed, 
we  should  have  log  (f>(w  +  A)  =  log  <fi(w)  ±  2  ttj.]  For  the  same 
reason  the  integral  which  is  taken  along  the  straight  line  leading 
from  ic  -\-  do  to  -^  c  +  Ais  also  zero.  But  along  the  two  lines 
which  bound  the  strip  from  to  to  to  -\-  c,  and  from  w-\-Ato 
tjo  +  c  +  A,  log  (fi(io)  passes  through  the  same  values,  and  since 
these  lines  are  described  in  opposite  directions,  the  integrals 
taken  along  them  cancel  each  other.  Consequently  the  inte- 
gral in  the  preceding  equation,  to  be  taken  along  the  entire 
boundary  of  the  parallelogram,  is  equal  to  zero,  and  therefore 

n  =:  1. 

Hence  the  function  <fi{to)  becomes  zero  only  once  in  the  strip 
considered.  But  then  it  can  also  acquire  any  arbitrary  value 
k  only  once  in  the  same  strip ;  for,  if  we  form  the  function 
<f>(tjo)  —  k,  this  is  periodic  just  as  <f>(tv)  is  periodic,  and  it  becomes 
infinite  only  once  for  to  =  r  just  as  <i>(tJo)  does ;  therefore  it  also 
becomes  zero  only  once  in  the  same  strip,  i.e.,  <f>{to)  becomes 
equal  to  k  only  once. 

We  can  now,  by  §  29,  let 
(1)  ^  =  <^(^,)=_^4.^(^„), 


264  THEORY  OF  FUNCTIONS. 

wherein  c  denotes  a  given  constant,  and  i/'(i«)  a  function  which 
no  longer  becomes  infinite  in  the  strip  to  be  considered,  but 
only  in  the  other  strips.     From  this  follows 

(2)  £=^.('«)=-(^.+^'W. 

Since  now  ^'(tu)  remains  finite  everywhere  in  the  strip,  therefore 

dz 

—  also  becomes  infinite  only  for  to  =  r,  i.e.,  only  where  z  be- 

dw 

comes  infinite,  and  this  result  must  hold  in  like  manner  for 

fly 

all  the  strips.     But  while  z  is  infinite  of  the  first  order,  —   is 

dz         ^^ 
infinite  of  the  second  order.    Hence,  if  we  regard  —  as  a  func- 

dw 

tion  of  z,  it  is  infinite  only  for  2  =  cc,  and  then  of  the  second 
order.  Since,  moreover,  z  acquires  each  value  only  once  in 
one  and  the  same  strip,  there  corresponds  only  one  value  of  w 
to  each  value  of  z,  in  one  and  the  same  strip.  Consequently  w 
is  a  function  of  z  which  has  indeed  an  infinite  number  of  values 
for  each  value  of  2,  but  these  values  differ  from  one  another 
only  by  multiples  of  the  modulus  of  periodicity,  i.e.,  by  constant 

quantities.     Accordingly  —  is  a  uniform  function  of  z,  since 

dz 

the  constants  vanish  in  the  differentiation.   Hence  the  reciprocal 

dz 
function  —  must  likewise  be  a  uniform  function  of  z.     If  we 
dw  , 

combine  this  with  the  preceding  results,  it  follows  that  —  is 

dio 
a  uniform  function  of  z,  which  becomes  infinite  only  for  z  =  oc, 

dz 
and  here  of  the  second  order.     Consequently  —  is  an  integral 

dtv 

function  of  z  of  the  second  degree  (§  31).  Such  a  function 
must  by  §  36  also  twice  acquire  the  value  zero.  If  we  denote 
by  a  and  6  the  values  of  z  for  which  this  occurs,  and  by  O  a 
constant,  we  have 

(3)     .  ^=C(z-a)(z-b), 

dw 
and  hence 


=/; 


dz 

C(z  -a)(z-  b)' 


MODULI  OF  PERIODICITY. 


265 


Therefore  a  simply  periodic  function,  whicli  becomes  infinite 
of  the  first  order  only  for  one  finite  point  in  each  strip,  is  the 
inverse  function  of  the  preceding  algebraic  integral. 

The  quantities  a  and  b  cannot  have  equal  values  in  this 
integral,  for  in  that  case  the  function 


J  Clz-d 


C{z  -  ay 

would  be  a  uniform  function  of  the  upper  limit  (§  60),  and 
then  z  could  not  be  a  periodic  function. 

The  constant  O  can  be  expressed  in  terms  of  c ;  for  from  (3) 
we  get 


C  =  lim 


dz 
dw 


[z^ 


and  with  help  of  equations  (1)  and  (2) 


C'  =  lim 


(lo  —  r) 


-,+  ^'(^) 


(w  —  r  )    _J»=r 


or 


We  then  have 


w 


=/; 


—  cdz 


(z  —  a){z  —  b) 


The  modulus  of  periodicity  A  is  equal  to  the  value  of  this 
integral,  taken  along  a  closed  line  which  encloses  either  the 
point  a  or  the  point  b.  If  we  integrate  round  a  in  the  direction 
of  increasing  angles,  we  obtain 


A  =  2  TTi  lim 


—  c{z  —  g)   ~[      _  2  Trie  , 
Jz-a)(z-b)\,^^      b-a 


for  integration  round  b  we  should  obtain  the  opposite  value. 
If  we  assign  the  value  Ti  to  the  lower  limit,  i.e.,  if  z  acquire  the 


266 


THEOBY  OF  FUNCTIONS. 


value  h  at  the  point  w  =  0,  we  have,  since  for  w  =  r  and  w  =  s, 
z  —  cc  and  z  =  0  respectively, 

J^°°        —  cdz  _  r"       4-  cdz 

h    (z~  d)(z  —  b)'  Jo  (2  —  a)(z  — 


(z  —  a)(z—  b) 


3.    T/ie  Inverse  Sine. 


vr 

Here  the  ^-surface  for  the  function 

1 


m  = 


Vl-z' 

consists  of  two  sheets.  We  have  the  two  branch-points  z  =  + 1 
and  z  =  —  l,  which  are  at  the  same  time  points  of  discontinuity. 
But  these  points  need  not  be  excluded,  since  f(z)  becomes  in- 
finite at  them  only  of  the  order  i ;  on  the  other  hand  the  point 
z  =  cc  must  be  excluded,  because 

limr-  '    ^       1 


VVl  —  zy^'^     V—  1 
is  finite,  and  in  fact  the  point  oo  must  be  excluded  in  both 
sheets,  since  it  is  not  a  branch-point.      For  this  reason  the 

connection  of  the  surface  in 
this  example  remains  the 
same,  whether  we  assume 
that  the  two  sheets  of  the 
z-surface  are  closed  at  infin- 
ity, or  imagine  a  closed  line 
drawn  in  each  sheet  as  a 
boundary,  and  then  enlarge 
these  lines  indefinitely.  In 
Fig.  63  the  latter  mode  of 
representation  is  chosen  on 
account  of  its  greater  prac- 
ticability. The  branch-cut 
is  drawn  from  —  1  to  -f- 1, 


MODULI  OF  PERIODICITY.  267 

and  the  lines  running  in  the  second  sheet  are  dotted.  This 
surface,  T,  is  doubly  connected,  and  the  cross-cut,  in  order  not 
to  divide  the  surface,  must  cross  the  branch-cut.  It  is  denoted 
by  the  line  ode,  the  part  dc  of  which  runs  in  the  second  sheet. 
The  modulus  of  periodicity  is  the  value  of  the  integral 


J 


dz 


vT 


taken  in  the  direction  of  increasing  angles  along  a  closed  line 
which  encloses  the  two  points  —  1  and  -f  1 ;  this  line  may  be 
drawn  either  in  the  first  or  in  the  second  sheet.  If  we  assume 
that  the  positive  sign  is  to  be  attached  to  the  radical  at  the 
points  which  lie  in  the  first  sheet  in  the  immediate  vicinity  of  '^ 
the  branch-cut,  and  on  the  left  side  of  the  same  taken  in  the  \ 
direction  from  —  1  to  4-1)  and  if  we  let  the  closed  line  run  in 
the  first  sheet,  we  can  contract  this  line  up  to  the  branch-cut, 
and  we  then  have 

We  have  seen  (§  43)  that  we  can  determine  the  value  of  this 
integral  by  regarding  the  closed  line  as  a  line  which  encloses 
the  point  co,  and  consequently  we  obtain 


For  a  line  running  in  the  second  sheet  we  should  have  obtained 
the  value  -|-  2  tt  ;  and,  in  fact,  a  line  which  makes  a  circuit 
round  —  1  and  -f  1  in  the  second  sheet  in  the  direction  of 
increasing  angles  crosses  the  cross-cut  in  a  direction  opposite 
to  that  of  a  similar  line  in  the  first  sheet.  Hence  the  inverse 
function  sin  w  of  the  preceding  integral  is  periodic  with  the 
period  2  tt. 

In  order  to  determine  the  mode  of  representing  the  z-surface 
on  the  w-surface,  we  will  let  z  describe  the  entire  boundary  of 
T'  in  the  positive  direction,  beginning  at  a,  where  iv  has  a 
value  denoted  by  it\.     If  the  outer  boundary  situated  in  the 


? 


? 


268  THEOBT  OF  FUNCTIONS. 

first  sheet  be  described  by  the  variable  z,  then  iv  goes  from  w„ 
to  w„  —  2  IT  along  a  line  the  form  of  which  depends  upon  the 
form  of  the  boundary-line  in  z  (Fig.  64).  Kow  let  z  go  from  a 
to  c  along  the  left  edge  (directed  from  a  to  c)  of  the  cross-cut 
ac,  and  to  from  w„  —  2  tt  to  a  value  which  may  be  denoted  by 
w^.  The  line  along  which  w  moves  may  again  differ  in  form 
according  to  the  form  of  the  cross-cut  ac.  Let  z  next  describe 
from  c  the  outer  boundary  of  the  second  sheet ;  then  w  goes 
from  w^  to  tr^  +  2  TT  along  a  curve 
which  depends  only  upon  the  outer 
boundary  of  the  second  sheet  of  the 
2;-surface.  Finally  z  closes  its  circuit, 
by  returning  along  the  left  edge  (di- 
rected from  c  to  a)  of  the  cross-cut  ca 
to  the  initial  point ;  then  iv  also  re- 
turns from  w;<,  -f  2  TT  to  w„.  The  line 
l'vOo-^2n  along  which  vo  last  moves  must  be 
parallel  to  the  path  (w„  —  2  tt,  w<,)) 
because  these  two  lines  correspond  to 
the  two  edges  of  the  cross-cut,  and 
because  v^  has  values  which  differ  by  2  tt  at  every  pair  of  infi- 
nitely near  points  on  the  two  edges.  If  we  now  enlarge  indefi- 
nitely the  outer  boundaries  of  the  surface  T,  then  the  lines 
iyo^,  w,,  —  2  tt)  and  (w,,,  w^  +  2  tt)  move  away  to  infinity,  and  z, 
or  sin  w,  acquires  all  its  values  in  one  strip,  which  is  bounded 
by  the  parallel  lines  AB  and  CD.  But  in  such  a  strip  z 
acquires  all  its  values  twice ;  for,  since  the  2:-surf ace  consists  of 
two  sheets,  there  correspond  two  values  of  w  to  each  value  of 
z,  not  taking  into  consideration  the  modulus  of  periodicity,  and 
hence  z,  or  siniy,  acquires  the  same  value  at  two  different 
points  vo. 

If  we  assume  that  the  cross-cut  ac  runs  along  the  ordinate 
axis,  so  that  on  both  its  edges  z  =  iy  (where  y  is  real),  we 
obtain 

w  =  *  I    —  ; 


MODULI  OF  PERIOBICITT.  269 

thus  to  is  also  a  pure  imaginary  or  differs  from  a  pure  imaginary 
quantity  by  multiples  of  the  real  modulus  of  periodicity  2  ir. 
The  tc-plane  is  then  divided  into  strips 
by  parallel  straight  lines,  which  run 
parallel  to  the  ?/-axis  and  pass  through 
the  points  0,  ±  2  tt,  ±  4  tt,  etc. 

In  order  to  determine  the  relation     J[ 
between  two  points  lo  and  w'  in  the 
same  strip,  to  which  correspond  equal 
values  of  z,  Ave  let  the  latter  variable 
first  pass  from  the  point  0  in  the  first 

sheet  to  the  point  0'  in  the  second  sheet,  situated  immedi- 
ately below  0,  without  crossing  the  cross-cut.  This  is  done 
(Fig.  65)  by  passing  along  the  branch-cut  round  +1,  next  along 
the  other  side  of  the  same  and  then  across  the  branch-cut  into 
the  second  sheet.     On  this  path  we  obtain  at  0'  the  value 


Jo    Vl  —  z2     J 1    Vl  —  z2 


Consequently,  the  point  iv  =  -n-  corresponds  to  the  point  z  =  0' 
situated  in  the  second  sheet.  If  z  now  go  from  0  to  z  in  the 
first  z-sheet,  iv  goes  from  0  to  lo.  But  if  z  go  in  the  second 
sheet  from  0'  to  z',  where  z'  is  situated  immediately  below  z, 
then  w  starts  with  the  value  tt,  and  because  the  radical  Vl  —  z^ 
has  the  negative  sign  in  this  part,  it  acquires  at  z'  the  value 


r 


dz 


0   Vl  —  z 


J"*"      dz 
0     A/f^T" 


but  w      _       __ 

Vl  —  z' 
and  consequently 

tV  +  ZV'  =   TT, 

or  the  sum  of  the  two  values  of  w,  for  which  z,  or  sintv, 
acquires  the  same  value,  is  equal  to  half  the  modulus  of  perio- 
dicity, not  taking  into  consideration  multiples  of  the  latter. 


270  THEORY  OF  FUNCTIONS. 

4.    The  Elliptic  Integral, 
dz 


=i: 


V(l  -  z^)  (1  -  A;V) 


Here  the  2-surface  consists  likewise  of  two  sheets,  and  has 

the  four  discontinuity-  and  branch-points  -f  1,  —  1,  -|-  -, 

k       k 

None  of  these  points  need  be  excluded,  because  the  function 
under  the  integral  sign  becomes  infinite  at  each  of  them  only 
of  the  order  i     The  point  oo  also  need  not  be  excluded,  since 


lim  lzf{z);\,^  =  lim 


_ V(l  -  z^)  (1  -  A;V)_ 


0. 


Consequently,  in  this  case  no  point  need  be  excluded.  This 
is  in  conformity  with  the  condition  that  the  preceding  in- 
tegral, as  we  have  already  seen  (§  45),  remains  finite  for  every 
value  of  z,  and  hence  can  become  infinite  only  by  the  addition 
of  an  infinitely  great  multiple  of  a  modulus  of  periodicity.  If 
we  assume  that  the  2;-surface  is  closed  at  infinity,  we  have  to 
do  with  a  surface  which  is  not  bounded  at  all  (or  only  by  an 
arbitrary  point),  but  which  is  multiply  connected.  In  such  a 
surface  we  let  the  first  cross-cut  be  a  line  returning  into  itself 
(§  47).     If  we  assume  that  the  points  —  1  and  -f  1  on  the  one 

hand,  and  -1-  -  and  —  -  on  the  other,  are  connected  by  branch- 

k  k 

cuts,^  we  will  take  for  the  first  cross-cut  a  line  q^,  which 
encloses  the  two  points  —  1  and  + 1  in  the  upper  sheet 
(Fig.  66).  Such  a  line  does  not  divide  the  surface,  since  we 
can  pass  from  one  side  to  the  other  side  of  the  same.  The 
way  in  which  this  passage  is  made  (cf.  §  46,  v.)  indicates 
how  the  second   cross-cut  ^2  is  to  be   drawn;   namely,  from 

1  In  Fig.  66  it  has  been  likewise  assumed  that  k  is  real  and  less  than 

unity;   then  the  branch-cut  drawn  from  +-  to  --  passes  through  cc. 

k  k 

But  we  will  first  consider  k  as  &  quite  arbitrary  quantity,  and  only  later 
return  to  the  assumption  that  k  is  real  and  less  than  unity. 


MODULI  OF  PERIODICITY.  271 

a  point  a  of  the  first  cross-cut  across  the  branch-cut  (—  1,  -f  1) 
into  the  second  sheet,  then  across  the  other  branch-cut  back 
again  into  the  first  sheet,  returning  in  this  sheet  to  the  initial 
point,  but  on  the  other  side  of  the  first  cross-cut  (to  a'").  These 
two  lines  now  form  together  a  continuous  path,  in  which  each 


of  the  two  cross-cuts  is  described  twice  in  opposite  directions. 
The  arrows  indicate  this  description  in  the  positive  direction. 
In  this  surface  T'  every  closed  line  forms  by  itself  alone  the 
complete  boundary  of  a  portion  of  the  surface,  and  hence  the 
surface  is  simply  connected.  Its  boundary  is  formed  by  V 
the  two  edges  of  the  cross-cuts.  Thus  the  original  surface 
was  triply  connected. 

The  modulus  of  periodicity  Ai  for  the  cross-cut  qi  is  the 

integral  |  clw,  taken  in  the  direction  of  increasing  angles 
along  a  closed  line  which  leads  from  one  side  of  the  cross-cut 
to  the  other  side  of  the  same,  e.g.,  along  Qz-  This  line  can  be 
contracted  until  it  coincides  with  two  straight  lines,  one  of 

which  leads  from  -  to  1  in  the  first  sheet,  the  other  from  1  to  - 
k  tc 

in  the  second  sheet.  If  we  then  assume  that  the  sign  -f  is  to 
be  attached  to  the  radical  in  the  first  sheet,  and  if  for  brevity 
we  let 


V(l  -  z^)  (1  -  Jrz")  =  A(z,  k), 
we  have 

^     Ji  A(z,  k)     Ji  A(z,  k)  Ji  A(z,  k) 


272 


THEORY  OF  FUNCTIONS. 


^ 


The  modulus  of  periodicity  A2  for  the  second  cross-cut  is  in 
like  manner  equal  to  the  integral  taken  along  the  line  qi  and 
this  line,  as  in  the  former  case,  can  be  contracted  up  to  the 
branch-cut ;  then,  as  before, 

J+i  A{z,  k)     J -I  A{z,  k)  J-\  A(z, 

or  also,  as  is  evident, 

•Jo    . 


ky 


dz 

^0  A{z,  k) 


The  elliptic  integral  therefore  has  two  different  moduli  of 
periodicity;  consequently  the  inverse  function,  the  so-called 
elliptic  function,  which  is  designated  after  Jacobi  by  sin  am  w, 
is  doubly  periodic. 

If  we  now  represent  the  z-surface  on  the  w-surface,  we 
obtain  the  following  results  :  If  2;  go  from  a  along  the  cross- 
cut Qi  in  the  direction  of  increasing  angles  and  at  the  same 
time  in  the  positive  boundary-direction,  and  therefore  return 
to  a  on  the  inner  edge  of  the  line  gj  (in  Fig.  66  from  a  to  a'), 
then  w  increases  from  ^o  to  m;  -f  A2.  In  this  tv  passes  along  a 
line  (Fig.  67)  the  form  of  which  depends  upon  the  form  of  the 
line  Qi  (to  be  chosen  arbitrarily) ;  if  z  next  go  along  the  line 

92  in  the  same  direction  to  a 
again  (i.e.,  from  a'  to  a"),  w 
increases  from  to  +  A2  to 
10  +  Ai  -\-  A.,,  along  a  line 
which  changes  its  form  with 
that  of  52-  If  ^  then  de- 
scribe the  line  cji  starting 
from  a",  always  in  the  posi- 
tive boundary-direction,  but 
now  in  the  direction  of  decreasing  angles  (i.e.,  from  a"  to 
a'"),  w  goes  from  w  -{-  Aj  -\-  A2  to  w  +  A^,  because  it  is  dimin- 
ished by  A^.  The  line  along  which  this  movement  of  w  takes 
place  must  be  parallel  to  the  line  (w,  w  +  A^),  because  the  two 
values  of  w  at  every  two  infinitely  near  points  on  the  two  edges 


W+SAy+Ai 


Fig.  07. 


MODULI  OF  PEBIODICITY.  273 

of  the  line  Qj  differ  by  the  quantity  Ai,  and  hence  two  different 
but  parallel  lines  in  w  correspond  to  the  two  edges  of  this 
cross-cut.  Finally,  if  z  go  from  a'"  to  a  along  the  cross-cut 
g,,  then  w  goes  from  w  +Ai  to  w  along  a  line  which  for  the 
same  reason  as  before  must  be  parallel  to  the  line  (w  -f  A2, 
w  -{-  Ai-}-  A2).  Thus  to  the  two  edges  of  the  cross-cut  Qi  cor- 
respond the  parallel  lines  (w,  w+A^)  and  (lo  -f  Ai,  w+Ai+Az), 
and  to  the  two  edges  of  the  cross-cut  ^2  the  parallel  lines 
(w,  to  -f  ^1)  and  (w  -f  Ao,  w  -\-  Ai-{-  A2).  Now  to  all  the 
points  z  in  the  whole  infinite  extent  of  the  z-surface  correspond 
only  such  points  w  as  lie  within  ^  the  curvilinear  bounded 
parallelogram,  for  a  line  can  be  drawn  through  any  arbitrary 
point  of  the  z-surface  which  leads  from  one  side  of  q^  to  the 
other  side  of  gi,  without  crossing  a  cross-cut;  hence  the  cor- 
responding line  tv  leads  from  the  line  (w,  w  +  A2)  through  the 
interior  of  the  parallelogram  to  the  line  (w  -f  Ai,  w  +  Ai  +  A2). 
Consequently  z,  or  sin  am  iv,  acquires  all  its  values  in  this 
parallelogram,  and  indeed  each  value  twice,  since  the  ^-surface 
consists  of  two  sheets. 

Other  parallelograms  now  adjoin  this  parallelogram  on  all 
sides.  For  if  we  let  z  go  from  a  to  a'",  for  instance,  then  w 
goes  from  w  to  w  -\-  Ai.  But  if  we  now  let  w  proceed  continu- 
ously across  the  cross-cut  Qi,  then  w  starts  with  the  value  w+Ai; 
hence  to  the  side  {tv  -f-  Ai,  w  -\-  Ai  +  A2)  is  joined  a  new  paral- 
lelogram, at  the  corners  of  which  iv  has  the  values 

w  -\-  Ai,  w  -^  Ai  +  A2,  to  -\-  2Ai  +  A2,  w  -\-2Ai. 

Similarly  for  the  three  other  sides.  In  this  way  the  whole 
?(;-plane  is  divided  into  parallelograms  by  two  sets  of  parallel 
lines.     If  we  assume  that  k  is  real  and  less  than  unity,  the  four 

points  -j-  1,  —  1,  H — , lie  on  the  principal  axis ;  if  we  now 

k       k 

contract  the  two  cross-cuts,  so  that  they  run  along  the  two 
edges  of  the  principal  axis,  the  parallel  lines  become  straight 
lines,  which  run  parallel  to  the  x-  and  the  y-axis  respectively. 

1  Within,  because  w  remains  finite  for  all  values  of  z. 


274  THEORY  OF  FUNCTIONS. 

J'»i  dz 

is  real.     We  usually  designate  the  value  of  this  integral  after 
Jacobi  by  K.     The  other  integral 
1 


i 


dz 


1  V(l-z')  (1-^:^2^; 


on  the  other  hand,  is  a  pure  imaginary.    If  we  let  Vl  —  k^  =  Jc' 
and  transform  the  integral  by  the  substitution 


k 
dz' 


^'^''     -'rvw^ 


V(l  -  z%l  -  k'^zy 

which  is  designated  by  —  ilP.  Consequently  the  moduli  of 
periodicity,  except  as  to  signs,  are 

4^and2i^'. 

We  can  in  this  example  also  determine  the  relation  between 
every  pair  of  values  of  w  which  correspond  to  the  same  value  of 
z;  i.e.,  to  two  points  of  the  z-surface  lying  one  immediately 
below  the  other.  To  the  value  z  =  0  in  the  first  sheet  cor- 
responds m;  =  0.  In  order  to  come  to  0'  in  the  second  sheet,  we 
can  conceive  the  cross-cut  q2  to  be  so  enlarged  that  it  also 

encloses  the  origin  as  well  as  the  points  1  and  —    We  can  pass 

k 

within  T'  from  0  along  the  branch-cut  rovmd  the  point  -f  1 
to  the  other  side  of  the  branch-cut  and  then  across  the  same 
to  0'  (cf.  p.  269) ;  w  then  acquires  at  0'  the  value 


Jo  A(z,k)     Ji  A(z,k) 


and  is  therefore  equal  to  the  half  of  one  of  the  moduli  of 
periodicity.      If  z  now  go  from  0'  to  z',  where  z'  lies  in  the 


MODULI  OF  PERIODICITY.  275 

second  sheet  immediately  below  z,  we  have,  designating  the 
value  of  w  at  z'  by  w', 

Jo  A(z, , 


dz 


u  4-  C    dz 

but  w  =  I  —^ 

Jo  A(z, 

and,  therefore, 

w  +  w'  =  2  K 

If   we  take  the  integral  I  dw  along  a  closed  line  which 

encloses  all  four  branch-points,  such  a  line  runs  entirely  in  the 
first  sheet  (Fig.  66),  and  hence  forms  by  itself  alone  a  com- 


FiG.  66. 

plete  boundary.  Consequently,  this  integral  has  the  value 
zero.  If  we  now  contract  this  line  up  to  the  principal  axis, 
on  which  are  the  four  branch -points,  the  integral  is  divided 
into  the  following  parts  (the  lines  may  be  described  in  the 
direction  of  decreasing  angles) : 

(1)  from  -  1  to  +1 ;  ^^^  f^^^  _  1  ^j^^^^gj^  00  to  +^; 

(2)from+lto+i;  ^ 

(o)  from  +  -  to  +  1 ; 

(3)  from  +  -  through  co  to : 

^  ^  k  ^  ^-'  (6)  from  +  1  to  - 1. 

The  radical  is  to  be  taken  negatively  in  (6)  and  (4),  because 
for  these  the  path  of  integration  lies  on  the  right  side  of  the 

branch-cuts  (—  1,  +  1)  and  (+-, );  in  all  the  others  it  is 

\        rC  Kj 


? 


276  THEORY  OF  FUNCTIONS. 

to  be  taken  positively.     Consequently  (2)  and  (5)  cancel  each 
other,  and  (1)  and  (3)  are  to  be  doubled.     Since,  further, 


(1)  =  2  C-4^—,  (3)  =  2  f 


dz 


we  obtain 


k 

Jo  A(2!,  A;)     Ji   A(z,  k)  ~   ' 


and  hence  i     .  ^'^,,  —  —  K. 


dz 

h    A  (2;,  A;) 


From  this   result  follows   also   the  value   of  the  integral 
between  the  limits  0  and  00  ;  for  since  this  is  divided  into  the 

parts  0  •••  1,  1 , Qo,  we  obtain 


A(2,  k) 


or,  since  we  can  add  to  this  value  the  modulus  of  periodicity 
2  iK\  also 


X 


A(2;,  k) 


Thus  z  becomes  infinite  within  the  parallelogram  with  the 
corners    0,    4  7f,   4  /iT  +  2  ilV,    and   2  iK'    for    w  =  I'/i"'    and 

We  will  also  in  this  example,  following  the  method  of 
Eiemann,  consider  the  relation  between  the  doubly  periodic 
function  and  the  elliptic  integral  in  the  inverse  manner,  i.e., 
starting  from  the  doubly  periodic  function.  Let  ^{w)  be  a 
uniform  doubly  periodic  function,  and  therefore  possess  the 
property  that  simultaneously 

^{w  +  A^  —  <l^(^)  and  <f)(w  -\-  A2)  =  ^{w). 

Then  the  straight  lines  which  represent  the  complex  quantities 
Ax  and  A^  must  have  different  directions.  For  if  they  have 
the  same  direction.  At,  and  A^  must  possess  a  real  ratio  (§  2,  3). 
This  can  be  either  rational  or  irrational.     If  it  be  rational, 


MODULI  OF  PERIODICITY.  277 

Ai  and  Ao,  are  commensurable,  and  hence  are  multiples  of  one 
and  the  same  quantity  B.     We  can  thus  let 

Ai  —  mB,  A2  =  nB, 

wherein  m  and  n  denote  two  integers,  which  are  relatively 
prime  to  each  other,  and  we  then  obtain 

<i>(w)  =  <f>(w  +  mB)  =  (f>(w  +  nB). 

Now  since  in  this  case  there  are  two  integers  a  and  b  connected 
by  the  relation 

ma  —  nb  =  1, 
and  since,  moreover, 

<fi(w  +  mxiB  —  nbB)  =  <f>(w), 

we  also  have  <l>(tv  +  B)=<f>(w), 

and  hence  in  this  case  the  function  <f>(w)  is  simply  and  not 
doubly  periodic.  But  if  Ai  and  A^  have  a  real  irrational  ratio, 
so  that  they  are  incommensurable,  there  are  always  two  integers 
m  and  n  for  which  the  modulus  of  mAi  +  nAi  becomes  less 
than  any  assignable  quantity.^     Since  now  also 

<f)(w  +  mAi  +  nAz)  =  <i>(w), 

1  If  we  let  — ?  =  a,  then,  according  to  the  assumption,  a  is  real  and 
irrational.     If  we  develop  the  absolute  value  |  a  |  of  a  in  a  continued  frac- 

u 

tion,  and  if  we  denote  two  consecutive  convergents  of  the  same  by  -  and 

/*'  " 

-ji  then,  as  is  well  known,  for  the  absolute  value 


{^|•|} 


<^ 


aj id  hence  (jit— k  |  a|)<— • 

v' 

But  since  the  denominator  of  the  convergents  increases  indefinitely,  we 
can  make  this  expression  as  small  as  we  please  by  continuing  the  develop- 
ment sufficiently  far.    But  we  have 

mAi  +  nAi  =  Ai(m  +  no) ; 

hence  if  we  let  m  =  fi  and  n  =  T  ",  according  as  |  a  |  =  ±  a,  we  can 
make  m  +  na,  and  therefore  also  the  modulus  of  mAi  +  71^2?  as  small  as 
we  please. 


278  THEORY  OF  FUNCTIONS. 

the  function  <f>{w)  maintains  the  same  value  for  an  indefinitely- 
small  change  of  the  variable,  and  hence  is  a  constant.  Conse- 
quently the  ratio  of  the  two  moduli  of  periodicity  of  a  doubly 
periodic  function  must  be  imaginary,  and  therefore  the  straight 
lines  Ai  and  A2  must  have  different  directions.  Then  we  can 
divide  the  w-plane  into  parallelograms  by  two  sets  of  parallel 
lines  in  such  a  way  that  <f>(w)  acquires  the  same  values  on  any 
two  parallel  lines ;  moreover,  it  then  acquires  all  its  values  in 
each  parallelogram,  and  has  the  same  value  at  every  two  corre- 
sponding points  of  different  parallelograms. 

Since  the  uniform  function  <f>(z)  must  become  infinite  for 
some  one  value  of  w  (§  28),  it  must  become  infinite  in  every 
parallelogram.  Let  us,  therefore,  select  any  parallelogram 
(Fig.  68),  and. let  r,  r',  r",  etc.,  be  the  points  of  the  same  at 
which  <l>(w)  becomes  infinite.     If  we  form  the  integral 


I  <f>(w)dw, 


taken  over  the  boundary  of  the  parallelogram,  then  by  §  19 
this  is  equal  to  the  sum  of  the  integrals  taken  round  the  points 
of  discontinuity  r,  r',  r",  etc.  Therefore,  if  (j>(w)  at  these 
points  become  infinite  in  the  same  way  as 

c       I   ...         c'       I   ...         c"       ,     ..   e4.„ 
1- ...     ; -!-•••?  r,  ^      ,610., 


respectively  do,  we  have 

<f>(w)dw  =  2  Tri(c  +  c'+  c"  -\ ). 


/■ 


But  <f>(w)  has  the  same  values  on  the  side  CE  as  on  DF,  the 
same  values  on  CD  as  on  EF,  and  in  the  description  of  the 
boundary  of  the  parallelogram  the  parallel  sides  are  described 
in  opposite  directions ;  hence  the  integrals  taken  along  these 
sides  cancel  each  other,  and  thus 

I  <l)(w)dw  =  0, 
consequently,  also      c-\-c'  +  c"-\ =0. 


MODULI  OF  PERIODICITY.  279 

From  this  we  conclude  that  <i>(tv)  must  become  infinite  more 
than    once    in    each    parallelo- 
gram, and  at  least  infinite  of 
the  first  order  at  two  points  or 

of    the    second    order    at   one  /   /^        (^ 

point.     If  n  denote  the  multi-  /       ^--^        ^"^ 

plicity  of  the  infinite  value  (or 

values)  of  di(iv)  in  each  paral-  „     „„ 

^  "^    „  Fio-  68. 

lelogram,  we  can  first  show  that 

<l>(w)  must  acquire  each  value  h  in  each  parallelogram  n  times. 

For  that  purpose  we  will  consider  the  integral 

taken  along  the  boundary  of  the  parallelogram.  This  also  has 
the  value  zero,  because  both  <f>(w)—  h  and  <f>'(w)  have  the  same 
values  on  the  opposite  sides  of  the  parallelogram.  But  on  the 
other  hand  this  integral  is  equal  to  the  sum  of  the  integrals 
taken  round  those  points  at  which  <^'(w)  becomes  infinite,  and 
round  those  at  which  (fi(iv)—h  vanishes.  The  former  are  the 
same  as  those  at  which  <l>(w)  or  (f)(iv)—  h  becomes  infinite  (§  29). 
Now  if  in  general  a  be  a  point  at  which  <i>{w)  —  h  becomes  either 
infinitesimal  or  infinite,  and  that  of  the  jpth  order  {p  positive 
for  infinitesimal  values),  we  can  put  (§  34) 

<i>{w)  —  h=(w  —  ay^(w), 

wherein  l/'(^o),  for  lu  =  a,  is  neither  zero  nor  infinite.  We  then 
obtain 

Therefore  i  d  log  [<^(^u)  —  A]  =  2  Trt'Sp, 

taken  round  the  entire  parallelogram,  and  hence 

Sp  =  0. 


280  THEORY  OF  FUNCTIONS. 

Now  <f>{w)—  h  becomes  n  times  infinite,  just  as  <)>(w)  does;  if 
m  denote  the  number  of  times  that  it  becomes  zero,  we  have 

2p  =  m  —n  =  0, 

and  hence  m  =  n. 

Since,  therefore,  <l>(w)  —  h  must  become  zero  n  times,  <i>(iv)  also 
becomes  n  times  equal  to  h. 

We  will  now  consider  in  the  following  only  the  simplest 
case,  in  which  <^(w)  becomes  infinite  twice  in  each  parallelogram 
and  therefore  also  acquires  every  value  twice.  We  will  first 
assume  that  ^(w)  becomes  infinite  of  the  first  order  at  two 
points  r  and  s.     Then,  denoting  <fi(w)  by  z,  we  can  put 

G  C 

z  =  <t>{w)= 1 h  if/(w), 

^^     to  —  r     w  —  s      ^  ^  ■" 

or,  since  c  +  c'  =  0, 

z  =  <l>{w)  =  — —  +  ip(w),  (4) 

wherein  c  denotes  a  given  constant,  and  il/(io)  a  function  which 
no  longer  becomes  infinite  in  the  parallelogram  under  consider- 
ation, and  therefore  only  in  the  other  parallelograms  at  the 
points  r  +  mAi  +  i^'^2  a^nd  s  +  mAi  +  nAs  (wherein  7n  and  n 
are  to  have  all  positive  and  negative  integral  values).  We  will 
first  determine  the  relation  between  the  two  values  of  w  for 
which  <t>(w)  has  the  same  value.     For  this  purpose  let 

V  =  r  +  s  —  w. 

If  we  substitute  v  for  w  in  (4),  we  get 

V  —  r     V  —  s 
But  since                             v  —  r  =  —  (w  —  s) 

V  —  s  =  —  (w  —  r), 

it  follows  that     <f>(v)  = 1 \-  Mv), 

w  —  s     w  —  r 

and  hence  fl>(^)—  <^(''^)  =  ^i}'^)—  ^{^)-  ■. 


MODULI  OF  PEEIODICITT.  281 

Therefore  this  difference  remains  finite  in  the  first  parallelogram. 
In  an  adjacent  parallelogram  cf>(tv)  becomes  infinite  &tw=r+Ai 
and  w  =  s  +  Ai;  hence  we  can  also  let 

w  —  r  —  Ai     to  —  s  —Ai     ^  ^  '" 

wherein  now  ^/'l(^«)  remains  finite  for  all  points  of  the  second 
parallelogram.     If  we  now  substitute 

Vi  =  r  +  s  +  2  Ai  —  w, 

we  get  w  —  r  —  Ai  =  —  (vi  —  s  —  Ai) 

w  —  s  —  Ai  =  —(vi  —  r  —  Ai), 
and  hence  also 

</'K)= ^^-^  +  — '-^—^ + -AiC^i) ; 

w  —  s  —  Ai     IV  —  r  —  Ai 

consequently  <f>(^f^)—  0(^0=  "AiC^O"  "Ai(''^i) 

and  remains  finite  within  the  second  parallelogram.  But  since 
Vi  differs  from  v  only  by  twice  the  modulus  of  periodicity  Ai, 
it  follows  that 

<f>(Vi)=  (}>{v),  <l>(iv)-  <f>ivi)=  <l>(w)-<j>(v)  ; 

hence  the  difference  </>(w)  —  <f>(v) 

remains  finite  in  the  second  as  well  as  in  the  first  parallelogram. 
If  we  continue  in  this  way  from  parallelogram  to  parallelo- 
gram, we  conclude  that  this  difference  does  not  become  infinite 
in  any  parallelogram  and  hence  not  at  all ;  therefore  it  must 
be  a  constant.     To  find  the  value  of  this  constant,  we  let 

r  +  s 


then 


2     ■ 
r  +  s 


2 

and  since  the  function  <f}  is  imiform,  also 
<^(v)=  <f>(w). 


282  THEORY  OF  FUNCTIONS. 

Therefore,  since  the  difference  <]>{w)  —  <j>{v)  has  the  value  zero 
for  one  value  of  to,  it  has  this  value  always,  and  hence 

<f>(r  -\-  s  —  w)=  <f>(w). 

Consequently  w  and  r  -\-  s  —  w  are  the  two  corresponding  values 
of  w  for  which  the  function  <f>(w)  acquires  the  same  value. 
From  (4)  it  follows  that 

dw  (to  —  ry      (tc  —  sy 

therefore,  not  taking  into  account  the  modidi  of  periodicity, 
the  derivative  <f>'(w)  is  infinite  only  for  w  =  r  and  w  =  s,  but 
for  these  it  is  infinite  of  the  second  order.  Hence  it  becomes 
infinite  four  times  in  every  parallelogram  and  therefore  also 
acquires  each  value  four  times.  It  is  likewise  a  uniform  func- 
tion of  w ;  but  it  is  important  to  inquire  whether  it  is  also  a 

dz 
uniform  function  of  z.     Kow  the  derivative  —   acquires   the 

dw 

same  value  at  every  pair  of  corresponding  points  of  different 
parallelograms  at  which  z  has  the  same  value.  Thus  we  have 
to  consider  only  the  points  v  and  w  of  the  same  parallelogram. 
If  we  differentiate  the  equation 


as  to  w,  we  obtain 


since 


Consequently  z  does  indeed  take  the  same  value  for  v  and  %, 

but  —  opposite  values;  therefore  —  is  not  a  uniform  func- 
dw  dw 

tion  of  z,  since  it  can  acquire  two  different  values  for  the  same 

value  of  z.      But  since  these  are  numerically  equal  and  of 

/'dz\^ 
opposite  signs,  it  follows  that  (  —  )    is  a  uniform  function 

dz  ^^^^ 

of  z.     Now  —  is  infinite  only  where  z  is  also  infinite,  but  it 

dw  ^  ' 

is  infinite  of  the  second  order  while  z  is  infinite  of  the  first 


<}>(w)=  . 

m 

<f>'(w)  =  - 

-<i>'(v), 

dv  _ 
dw 

-1. 

order ;    consequently 


MODULI  OF  PERIODICITY. 

dz 


283 


Therefore 


dz 
div 


die 


is   infinite   of   the  fourth  order. 


is  a  uniform  function  of  z,  which  becomes 


infinite  only  for  z  =  cc  and  that  of  the  fourth  order ;  accord- 
ingly it  is  an  integral  function  of  the  fourth  degree.  Such  a 
function  is  also  four  times  zero.  If  we  denote  by  a,  p,  y,  8, 
the  values  of  z  for  which  it  becomes  zero,  and  by  C  a  constant, 
we  have 


(5) 

from  this  is  obtained 


(£)'  =  ^^' "  "^^'  ~  ^^^'  -  y)(^  -  s) ; 


=/: 


dz 


^/C{z  -  a)(z  -  (3)iz  -  y)(z  -  8) 


Hence  a  doubly  periodic  function  which  becomes  twice  infinite 
of  the  first  order  in  every  parallelogram,  is  the  inverse  func- 
tion of  an  elliptic  integral.  The  constant  C  can  be  expressed 
in  terms  of  c.     For  since  by  (5) 


C=lim 


we  obtain 

C=lim 

=  lim 


dz 
dio 


l(w- 


ry     (w  —  sy 


\_w  —  r     IV  —  s  J 

r_  e  +  <^-^)'  +(w-  r) V'H 
[_  {w  —  sy 


b 


c{w  —  r) 
w  —  s 


+  (w  —  r)\p'{w) 


Then 


=/ 


cdz 


V(2-«)(2-/3)(2!-y)(^-8) 


284  THEORY  OF  FUNCTIONS. 

This  integral  admits  of  the  same  treatment  as  the  former 


/: 


dz 


V(l  -  z^)(l  -  Tc'z') 

if  we  put  the  four  branch-points,  a,  /8,  y,  8,  in  place  of  + 1, 

—  1,  -f  -, ;  it  can  also  be  transformed  into  the  latter. 

A;        k 

We  now  proceed  to  the  case  in  which  the  function  <^(w) 
becomes  infinite  only  at  one  point,  but  of  the  second  order. 
In  this  case  we  must  put 

(w  —  ry 

for  the  term  containing  {w  —  r)~^  must  be  wanting  in  order  that 

I  (f>(w)dw,  extended  over  the  boundary  of  the  parallelogram, 

may  have  the  value  zero.    We  infer  in  this  case,  just  as  before, 
that 

<f)(2r  —  w)=  <f>(w), 

by  letting  s  =  r,  and  hence 

<i>\2r-w)  =  -4>\io). 

Therefore  —  is  not  a  uniform  function  of  z,  but  again  ( - 
die  \c 


dz^^ 
vdwj 
is  a  uniform  function  of  z.     In  this  case 


dw  {w  —  ry 

dz 
thus  —  becomes  infinite,  of  the  third  order,  only  where  z  is 
dw  ^^ 

infinite  of  the  second  order.     Therefore  — ,  as  a  function  of  z, 

^^  rdzY 

is  infinite  of  the  order  f  for  z  =  cc,  and  consequently  I  — —  J 

is  infinite  of  the  third  order.     Accordingly  in  this  case  we  have 


MODULI  OF  PERIODICITY. 


285 


Therein 
C  =  liin 


div 


=  lim  < 


[■ 


^'    ,  +  ^'H^^ 


(w  —  r) 


[    [(^+^K 


consequently 


and 


-'V(2-«)C^-)8)(2-y) 

which  is  likewise  an  elliptic  integral. 

We  here  close  this  discussion,  because  it  is  not  the  purpose 
of  this  book  to  enter  more  in  detail  into  the  investigation  of 
periodic  functions;  but  the  cases  treated  are  to  be  regarded 
only  as  examples  illustrating  the  general  considerations. 


Supplementary  Note  to  Riemann's   Fundamental   Prop- 
osition ON  Multiply  Connected  Surfaces. 

Riemann  originally  gave  to  the  proposition  bearing  his  name 
(§  49)  a  somewhat  different  and  more  general  enunciation, 
which  presents  many  advantages,  while  it  removes  at  once 
a  difficulty  which  otherwise  requires  supplementary  exami- 
nation. 

This  differs  from  the  form  of  the  proposition  as  enunciated 
in  §  49  in  the  following  manner:  If  the  surface  T  be  first 
modified  by  q^  cross-cuts  of  a  first  mode  of  resolution  into 
a  system  Ti,  which  consists  of  «!  pieces,  and  a  second  time 
by  ^2  cross-cuts  of  a  second  mode*  of  resolution  into  a  system 
T^,  "which  consists  of  «2  pieces,  then  in  contradistinction  to 
the  enunciation  of  §  49  it  is  only  assumed  that  the  ai  pieces 
of  the  system  2\  are  all  simply  connected,  while  the  «2  pieces 
of  the  system  T^,  may  be  arbitrary ;  then  the  property  that 
^2  —  «2  cannot   be   greater  than   q^  —  ai  holds,   and  therefore 

g,  —  «2  ^  9i  —  «i- 

In  the  proof  of  this  property,  the  first  main  division  of  the 
proof  remains  exactly  the  same  as  in  §  49  or  §  51.  By  the 
superposition  of  the  two  systems  of  cross-cuts  a  new  system 
of  surfaces  ®  is  produced  in  two  ways,  and  it  is  proved  that 
if  the  lines  of  the  second  mode  of  resolution,  when  drawn  in 
Ti,  form  g'2-f  m  cross-cuts  in  that  surface,  then  the  lines  of 
the  first  mode  of  resolution,  when  drawn  in  T^,  also  consist 
of  gi  +  m  cross-cuts.  Since,  moreover,  Tj,  according  to  the 
hypothesis,  consists  of  Wj  simply  connected  pieces,  therefore 
Z  consists  of 

S  =  «i  +  g2  +  wi 
pieces. 

286 


SUPPLEMENTARY  NOTE.  287 

Now  the  system  Z  is  also  derived  from  T2,  which  consists 
of  ^2  pieces,  by  qi  +  m  cross-cuts.  Therefore  the  number  fS. 
of  pieces  of  which  gb  consists  can  (by  §  48,  V.,  note)  be  not 
greater  than  «£  +  5'i  +  "n,  but  on  the  contrary 

®  ^  «2  +  g'l  4-  m, 

i.e.,  ai  +  ^2  +  m^a2  +  gi  +  m; 

from  this  follows  immediately 

92  —  «2  ^  ^1  —  «!> 
which  was  to  be  proved. 

Therefore  g-j  —  «2  cannot  be  greater  than  Q'l  —  «! ;  and  if 
the  case  occur,  that  the  numbers  «i  and  ag  of  pieces  arising 
from  the  two  modes  of  resolution  are  equal  to  each  other, 
then  ^2  cannot  be  greater  than  qi. 

Conversely,  if  T2  consist  of  only  simply  connected  pieces 
(in  number  ag)?  while  the  Ui  pieces  which  form  the  system 
Ti  are  arbitrary,  we  have 

9i  —  «i  ^  ^2  —  az- 

But  if  both  systems  2^  and  T2  consist  of  only  simply  connected 
pieces,  then  g-j  —  «2  cannot  be  greater  than  q^  —  ud  nor  qi  —  ai 
be  greater  than  qz  —  oh'i  hence  in  this  case 

9i  —  «i  =  92  —  "2, 
and  this  is  the  principle  of  §  49. 

From  the  above  form  of  Kiemanu's  fundamental  proposition 
is  at  once  derived  the  second  proposition  of  §  52,  upon  which 
the  classification  of  surfaces  depends.  It  is  here  assumed  that 
a  multiply  connected  surface  T  can  be  changed  into  a  simply 
connected  surface  T^  by  q  cross-cuts  drawn  in  a  definite  manner, 
and  it  will  be  proved  that  this  modification  is  always  effected 
by  means  of  q  non-dividing  cross-cuts,  in  whatever  way  also 
the  latter  may  be  drawn.  From  the  above  proposition  follows, 
first,  that  the  surface  T  cannot  be  made  simply  connected  by 


288  THEORY  OF  FUNCTIONS. 

less  than  q  cross-cuts ;  hence  by  §  48,  II.,  it  is  possible  to  draw 
q  cross-cuts  in  such  a  definite  way  that  T  is  likewise  not 
divided.  Then  a  surface  T2,  which  consists  of  a  single  piece, 
again  arises.  But  this  cannot  be  multiply  connected ;  for  if  it 
were,  we  could  still  draw  in  it  at  least  one  non-dividing  cross- 
cut (§  48,  II.)  and  thus  obtain  by  ^  + 1  cross-cuts  a  surface 
consisting  of  one  piece,  while  the  simply  connected  surface  2^ 
arose  through  q  cross-cuts;  but  this  contradicts  the  above 
proposition  in  the  original  Riemann  form. 

The  property  proved  in  §  53,  V.,  also  requires  no  further 
proof  if  this  form  of  the  proposition  serve  as  the  basis,  but 
follows  at  once.  The  question  here  is  concerning  a  (g  +  l)-ply 
connected  surface  T,  which  is  therefore  made  simply  connected 
by  q  cross-cuts  and  is  divided  by  one  additional  cross-cut  into 
two  pieces.  If  a  dividing  cross-cut  B  be  first  drawn  instead  of 
these,  by  which  T  is  divided  into  two  pieces  A  and  B,  and  if  in 
these  pieces  additional  cross-cuts  be  drawn,  we  still  have  two 
pieces,  if  neither  A  nor  B  be  divided  by  the  new  cross-cuts. 
But  then  the  number  of  these  new  cross-cuts  possible  in  A  and 
B  cannot,  according  to  our  proposition,  be  greater  than  q,  and 
is  therefore  a  finite  number ;  from  this  the  remainder  follows, 
as  in  §  63,  V. 


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